[ Matrix Algebra ]
0 / 100 answered
10:00
Full-Length Practice Exam · Self-Study Edition

Matrix Algebra
From First Principles to the 3×3 Inverse

One hundred questions, arranged from the very first definition of a matrix up to solving a full 3×3 inverse — built so you can work through it alone, start to finish, and walk away having actually learned the unit.

100Questions
10Units
MCFormat
10:00Time Limit
minutes  ·  default is 10, change it to whatever fits your session
[
Unit 01 · Foundations

Matrix Basics

Notation, size, and elements — 10 questions
]
001 Easy
What is the size (dimensions) of this matrix?
251
0-34
Count rows first, then columns.
Why: The matrix has 2 rows and 3 columns, so its size is 2 x 3 (rows x columns).
002 Easy
In matrix
4-1
72
, what is the element in row 2, column 1?
Row number comes first, then column number.
Why: Row 2, column 1 is the entry in the second row, first column, which is 7.
003 Easy
A matrix with only one row is called a:
Think about what 'row' means here.
Why: A matrix with exactly one row (and any number of columns) is called a row matrix.
004 Easy
A matrix with the same number of rows and columns is called a:
'Square' shapes have equal sides.
Why: When the number of rows equals the number of columns, the matrix is a square matrix.
005 Easy
What is the size of
1
2
3
?
Rows go top to bottom.
Why: There are 3 rows and 1 column, so the size is 3 x 1.
006 Easy
Two matrices are equal only if they have the same size and:
Equality is checked entry by entry.
Why: Matrices are equal when their dimensions match and every corresponding entry is identical.
007 Easy
In matrix
9-20
, what is the element in row 1, column 3?
Column 3 is the last entry listed.
Why: Row 1, column 3 refers to the third entry in the single row, which is 0.
008 Easy
A matrix in which every entry is 0 is called the:
The name describes the value of every entry.
Why: A matrix whose entries are all 0 is called the zero matrix.
009 Easy
What is the size of a matrix with 4 rows and 2 columns?
The order is always rows x columns.
Why: By convention, size is written rows x columns, so 4 rows and 2 columns gives 4 x 2.
010 Easy
If matrix A has size 3 x 5, how many total entries does it have?
Multiply rows by columns.
Why: Total entries = rows x columns = 3 x 5 = 15.
[
Unit 02 · Foundations

Addition & Subtraction

Combining matrices entry by entry — 10 questions
]
011 Easy
Compute:
-5-5
0-5
 +
-1-1
3-2
Add or subtract entries that are in the same position.
Why: Add each pair of corresponding entries directly; matrices of the same size combine entry by entry.
012 Easy
Compute:
6-6
51
 +
2-5
0-5
Add or subtract entries that are in the same position.
Why: Add each pair of corresponding entries directly; matrices of the same size combine entry by entry.
013 Easy
Compute:
2-2
43
 +
-13
-35
Add or subtract entries that are in the same position.
Why: Add each pair of corresponding entries directly; matrices of the same size combine entry by entry.
014 Easy
Compute:
-5-6
4-3
 +
6-2
-5-3
Add or subtract entries that are in the same position.
Why: Add each pair of corresponding entries directly; matrices of the same size combine entry by entry.
015 Easy
Compute:
-50
-21
 -
4-1
-4-1
Subtract entries in the same position: top-left minus top-left, and so on.
Why: Subtract each pair of corresponding entries directly; matrices of the same size combine entry by entry.
016 Easy
Compute:
-1-3
4-2
 -
54
4-5
Subtract entries in the same position: top-left minus top-left, and so on.
Why: Subtract each pair of corresponding entries directly; matrices of the same size combine entry by entry.
017 Easy
Compute:
34
-42
 -
5-3
-41
Subtract entries in the same position: top-left minus top-left, and so on.
Why: Subtract each pair of corresponding entries directly; matrices of the same size combine entry by entry.
018 Easy–Med
Compute:
0-24
52-3
4-16
 +
6-6-3
-66-1
0-2-5
Add or subtract entries that are in the same position.
Why: Add each pair of corresponding entries directly; matrices of the same size combine entry by entry.
019 Easy–Med
Compute:
-335
-1-34
104
 +
1-4-2
-4-35
22-2
Add or subtract entries that are in the same position.
Why: Add each pair of corresponding entries directly; matrices of the same size combine entry by entry.
020 Easy–Med
Compute:
530
30-1
-3-42
 -
1-56
-6-5-4
4-46
Subtract entries in the same position: top-left minus top-left, and so on.
Why: Subtract each pair of corresponding entries directly; matrices of the same size combine entry by entry.
[
Unit 03 · Foundations

Scalar Multiplication

Multiplying a matrix by a constant — 8 questions
]
021 Easy
Compute 3 x
40
3-5
Multiply every single entry by the scalar.
Why: Scalar multiplication multiplies each entry of the matrix by 3.
022 Easy
Compute -2 x
00
31
Multiply every single entry by the scalar.
Why: Scalar multiplication multiplies each entry of the matrix by -2.
023 Easy
Compute 3 x
2-2
2-6
Multiply every single entry by the scalar.
Why: Scalar multiplication multiplies each entry of the matrix by 3.
024 Easy
Compute -2 x
45
-54
Multiply every single entry by the scalar.
Why: Scalar multiplication multiplies each entry of the matrix by -2.
025 Easy
Compute 3 x
26
-26
Multiply every single entry by the scalar.
Why: Scalar multiplication multiplies each entry of the matrix by 3.
026 Easy–Med
Compute 3 x
4-1-5
-20-4
Multiply every single entry by the scalar.
Why: Scalar multiplication multiplies each entry of the matrix by 3.
027 Easy–Med
Compute 4 x
1-6
55
-22
Multiply every single entry by the scalar.
Why: Scalar multiplication multiplies each entry of the matrix by 4.
028 Easy–Med
Compute 3 x
6-42
-54-2
Multiply every single entry by the scalar.
Why: Scalar multiplication multiplies each entry of the matrix by 3.
[
Unit 04 · Core Skills

Matrix Multiplication

Row-by-column products — 15 questions
]
029 Easy–Med
Matrix A is size 2 x 3 and matrix B is size 3 x 2. Is the product AB defined, and if so what size is it?
AB is defined when the number of columns of A equals the number of rows of B.
Why: A is 2x3 and B is 3x2: inner numbers (3 and 3) match, so AB is defined, and the result takes the outer dimensions: 2 x 2.
030 Easy–Med
Multiply:
32
11
 x
-2-2
-13
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
031 Easy–Med
Multiply:
-21
31
 x
-31
-10
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
032 Easy–Med
Multiply:
-3-3
-13
 x
3-1
-2-3
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
033 Easy–Med
Multiply:
-21
-3-3
 x
20
3-3
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
034 Easy–Med
Multiply:
31
3-2
 x
-22
01
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
035 Easy–Med
Multiply:
-20
42
 x
-1
4
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
036 Easy–Med
Multiply:
-10
21
 x
3
4
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
037 Easy–Med
Multiply:
3-3
-1-1
 x
-3
1
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
038 Medium
Multiply:
-31
 x
1-2
1-2
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
039 Medium
Multiply:
-3-3
 x
22
-3-2
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
040 Medium
Multiply:
-3-3
 x
3-1
-31
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
041 Medium
Multiply:
-2-12
0-21
 x
-22
11
0-2
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
042 Medium
Multiply:
30
30
-2-3
 x
-320
-100
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
043 Medium
Multiply:
13-2
333
 x
-2-21
30-2
-1-1-1
Each entry of the result is the dot product of a row from the first matrix and a column from the second.
Why: Multiply each row of the first matrix by each column of the second, summing the products, to fill in the result matrix.
[
Unit 05 · Core Skills

Transpose

Flipping rows and columns — 8 questions
]
044 Easy
Find the transpose of
21
-40
The transpose turns rows into columns.
Why: To transpose a matrix, write its rows as columns (row 1 becomes column 1, and so on).
045 Easy
Find the transpose of
-4-2
1-3
The transpose turns rows into columns.
Why: To transpose a matrix, write its rows as columns (row 1 becomes column 1, and so on).
046 Easy
Find the transpose of
-51
62
The transpose turns rows into columns.
Why: To transpose a matrix, write its rows as columns (row 1 becomes column 1, and so on).
047 Easy–Med
Find the transpose of
-5-64
2-6-5
The transpose turns rows into columns.
Why: To transpose a matrix, write its rows as columns (row 1 becomes column 1, and so on).
048 Easy–Med
Find the transpose of
6-3-4
011
The transpose turns rows into columns.
Why: To transpose a matrix, write its rows as columns (row 1 becomes column 1, and so on).
049 Easy–Med
Find the transpose of
-30
-6-4
0-6
The transpose turns rows into columns.
Why: To transpose a matrix, write its rows as columns (row 1 becomes column 1, and so on).
050 Easy
If matrix A has size 3 x 5, what is the size of A^T (the transpose of A)?
Transposing swaps the number of rows and columns.
Why: Transposing a matrix swaps rows and columns, so a 3 x 5 matrix becomes 5 x 3.
051 Easy–Med
For any matrix A, what is (A^T)^T equal to?
Think about transposing twice.
Why: Taking the transpose of a matrix twice returns the original matrix: (A^T)^T = A.
[
Unit 06 · Core Skills

Identity & Zero Matrices

Special matrices and their properties — 7 questions
]
052 Easy
What is
3-2
05
 x
10
01
 (multiplying by the identity matrix)?
Multiplying any matrix by the identity matrix changes nothing.
Why: The identity matrix I acts like the number 1 in multiplication: for any matrix A, A x I = A.
053 Easy
What are the entries on the main diagonal of a 3 x 3 identity matrix?
The identity matrix looks like a diagonal line of a single value.
Why: In an identity matrix, every entry on the main diagonal is 1, and all other entries are 0.
054 Easy
What is the result of adding any matrix A to the zero matrix of the same size?
Adding zero should not change anything.
Why: Adding the zero matrix leaves any matrix unchanged, just like adding 0 to a number.
055 Easy–Med
Which of these is the 2 x 2 identity matrix?
Diagonal entries are 1, everything else is 0.
Why: The identity matrix has 1s down the main diagonal and 0s everywhere else.
056 Easy–Med
For a square matrix A, what does A x I represent compared to I x A?
The identity matrix commutes with every square matrix.
Why: The identity matrix commutes with any square matrix of matching size, so A x I = I x A = A.
057 Easy–Med
What is the trace (sum of main diagonal entries) of a 3 x 3 identity matrix?
Add the three diagonal 1s.
Why: The main diagonal of a 3 x 3 identity matrix has three 1s, so the trace is 1+1+1 = 3.
058 Medium
If A x B = A x I for a square matrix A with an inverse, what must B equal?
Since A has an inverse, you can cancel A from both sides.
Why: Because A is invertible, multiply both sides by A^-1 on the left: B = I.
[
Unit 07 · Advanced

Determinant (2×2)

ad − bc, and invertibility — 10 questions
]
059 Easy
Find the determinant of
1-1
2-1
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
Why: det = (1)(-1) - (-1)(2) = -1 - -2 = 1.
060 Easy
Find the determinant of
13
52
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
Why: det = (1)(2) - (3)(5) = 2 - 15 = -13.
061 Easy
Find the determinant of
-3-2
-1-2
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
Why: det = (-3)(-2) - (-2)(-1) = 6 - 2 = 4.
062 Easy
Find the determinant of
-54
3-5
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
Why: det = (-5)(-5) - (4)(3) = 25 - 12 = 13.
063 Easy
Find the determinant of
0-5
-54
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
Why: det = (0)(4) - (-5)(-5) = 0 - 25 = -25.
064 Easy–Med
Find the determinant of
78
8-3
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
Why: det = (7)(-3) - (8)(8) = -21 - 64 = -85.
065 Easy–Med
Find the determinant of
-78
-6-3
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
Why: det = (-7)(-3) - (8)(-6) = 21 - -48 = 69.
066 Easy–Med
Find the determinant of
-6-6
-14
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
Why: det = (-6)(4) - (-6)(-1) = -24 - 6 = -30.
067 Easy–Med
Find the determinant of
-5-1
-7-6
For a 2x2 matrix [[a,b],[c,d]], det = ad - bc.
Why: det = (-5)(-6) - (-1)(-7) = 30 - 7 = 23.
068 Easy–Med
A 2x2 matrix has no inverse exactly when its determinant equals:
Division by the determinant is required to find an inverse.
Why: Finding an inverse requires dividing by the determinant, so a matrix is not invertible exactly when det = 0.
[
Unit 08 · Advanced

Determinant (3×3)

Cofactor expansion — 12 questions
]
069 Medium
Find the determinant of
021
11-1
-1-22
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: 0(1×2--1×-2) - 2(1×2--1×-1) + 1(1×-2-1×-1) = -3.
070 Medium
Find the determinant of
2-1-2
-10-2
22-1
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: 2(0×-1--2×2) - -1(-1×-1--2×2) + -2(-1×2-0×2) = 17.
071 Medium
Find the determinant of
0-13
-3-30
11-3
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: 0(-3×-3-0×1) - -1(-3×-3-0×1) + 3(-3×1--3×1) = 9.
072 Medium
Find the determinant of
-31-2
1-1-2
-1-3-2
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: -3(-1×-2--2×-3) - 1(1×-2--2×-1) + -2(1×-3--1×-1) = 24.
073 Hard
Find the determinant of
0-3-2
0-3-3
4-20
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: 0(-3×0--3×-2) - -3(0×0--3×4) + -2(0×-2--3×4) = 12.
074 Hard
Find the determinant of
0-11
-104
30-4
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: 0(0×-4-4×0) - -1(-1×-4-4×3) + 1(-1×0-0×3) = -8.
075 Hard
Find the determinant of
-320
-4-41
-20-2
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: -3(-4×-2-1×0) - 2(-4×-2-1×-2) + 0(-4×0--4×-2) = -44.
076 Hard
Find the determinant of
-414
-22-2
-401
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: -4(2×1--2×0) - 1(-2×1--2×-4) + 4(-2×0-2×-4) = 34.
077 Hard
Find the determinant of
-41-1
-1-31
42-2
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: -4(-3×-2-1×2) - 1(-1×-2-1×4) + -1(-1×2--3×4) = -24.
078 Hard
Find the determinant of
-23-2
33-2
0-3-2
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: -2(3×-2--2×-3) - 3(3×-2--2×0) + -2(3×-3-3×0) = 60.
079 Hard
Find the determinant of
2-13
032
323
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: 2(3×3-2×2) - -1(0×3-2×3) + 3(0×2-3×3) = -23.
080 Hard
Find the determinant of
-2-1-2
32-3
03-3
Expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), using the standard a,b,c / d,e,f / g,h,i labels.
Why: Expanding along row 1: -2(2×-3--3×3) - -1(3×-3--3×0) + -2(3×3-2×0) = -33.
[
Unit 09 · Mastery

Inverse (2×2)

Adjugate over determinant — 12 questions
]
081 Easy–Med
Find the inverse of
0-1
-2-2
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = -2. Swap a,d and negate b,c to get the adjugate, then multiply by 1/-2 to get the inverse.
082 Easy–Med
Find the inverse of
10
01
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = 1. Swap a,d and negate b,c to get the adjugate, then multiply by 1/1 to get the inverse.
083 Easy–Med
Find the inverse of
-13
-15
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = -2. Swap a,d and negate b,c to get the adjugate, then multiply by 1/-2 to get the inverse.
084 Easy–Med
Find the inverse of
10
1-1
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = -1. Swap a,d and negate b,c to get the adjugate, then multiply by 1/-1 to get the inverse.
085 Easy–Med
Find the inverse of
-42
-32
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = -2. Swap a,d and negate b,c to get the adjugate, then multiply by 1/-2 to get the inverse.
086 Easy–Med
Find the inverse of
22
-1-2
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = -2. Swap a,d and negate b,c to get the adjugate, then multiply by 1/-2 to get the inverse.
087 Medium
Find the inverse of
-4-1
-2-1
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = 2. Swap a,d and negate b,c to get the adjugate, then multiply by 1/2 to get the inverse.
088 Medium
Find the inverse of
10
32
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = 2. Swap a,d and negate b,c to get the adjugate, then multiply by 1/2 to get the inverse.
089 Medium
Find the inverse of
-11
0-2
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = 2. Swap a,d and negate b,c to get the adjugate, then multiply by 1/2 to get the inverse.
090 Medium
Find the inverse of
20
01
Swap a and d, negate b and c, then divide every entry by the determinant.
Why: det = 2. Swap a,d and negate b,c to get the adjugate, then multiply by 1/2 to get the inverse.
091 Easy–Med
What is the first step in finding the inverse of a 2x2 matrix [[a,b],[c,d]]?
You need the determinant before you can divide by it.
Why: The inverse formula requires dividing by the determinant, so it must be calculated first.
092 Medium
If det(A) = 0 for a 2x2 matrix A, what can you conclude?
The inverse formula divides by the determinant.
Why: Since the inverse formula requires dividing by det(A), a determinant of 0 means A is not invertible.
[
Unit 10 · Mastery

Inverse (3×3)

Cofactors, adjugate, and division by det — 8 questions
]
093 Hard
Find the inverse of
-13-1
-211
22-3
Find the matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant.
Why: det(A) = -1. Build the cofactor matrix, transpose it (adjugate), then divide each entry by -1 to get A^-1.
094 Hard
Find the inverse of
0-13
1-13
-3-12
Find the matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant.
Why: det(A) = -1. Build the cofactor matrix, transpose it (adjugate), then divide each entry by -1 to get A^-1.
095 Hard
Find the inverse of
10-1
-113
100
Find the matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant.
Why: det(A) = 1. Build the cofactor matrix, transpose it (adjugate), then divide each entry by 1 to get A^-1.
096 Hard
Find the inverse of
223
031
-31-3
Find the matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant.
Why: det(A) = 1. Build the cofactor matrix, transpose it (adjugate), then divide each entry by 1 to get A^-1.
097 Hard
Find the inverse of
-3-11
-2-3-2
-112
Find the matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant.
Why: det(A) = 1. Build the cofactor matrix, transpose it (adjugate), then divide each entry by 1 to get A^-1.
098 Hardest
Find the inverse of
-1-30
-301
-1-20
Find the matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant.
Why: det(A) = 1. Build the cofactor matrix, transpose it (adjugate), then divide each entry by 1 to get A^-1.
099 Hardest
Find the inverse of
10-1
23-3
-322
Find the matrix of cofactors, transpose it to get the adjugate, then divide every entry by the determinant.
Why: det(A) = -1. Build the cofactor matrix, transpose it (adjugate), then divide each entry by -1 to get A^-1.
100 Hardest
What is the correct order of steps to find the inverse of a 3x3 matrix?
Remember: Cofactor -> Adjugate -> Divide.
Why: The standard method is: compute the matrix of cofactors, transpose it to form the adjugate, then divide every entry by the determinant.
Test Complete
0/100
You answered 0 of 100 questions.
0
Correct
0
Incorrect
0
Unanswered
0:00
Time Used
Review
Only the questions you answered incorrectly are shown below — skipped questions aren’t included.
Built for self-study · work through it once, review every miss, and it should stick.
Answer Key