01
Matrix Basics & Notation
⚡ Quick Memory Point
ROWS first, COLUMNS second — always!
DIMENSION = rows × columns
"RC Cola" → Rows × Columns
A 3×2 matrix = 3 rows, 2 columns (not the other way!)
Element \(a_{ij}\) → row i, column j
"RC Cola" → Rows × Columns
A 3×2 matrix = 3 rows, 2 columns (not the other way!)
Element \(a_{ij}\) → row i, column j
📐 Worked Example
Given \(A = \begin{bmatrix} 2 & -1 & 5 \\ 0 & 3 & 7 \end{bmatrix}\)
• Dimension of A = 2 × 3 (2 rows, 3 columns)
• Element \(a_{12}\) = −1 (row 1, column 2)
• Element \(a_{23}\) = 7 (row 2, column 3)
• Dimension of A = 2 × 3 (2 rows, 3 columns)
• Element \(a_{12}\) = −1 (row 1, column 2)
• Element \(a_{23}\) = 7 (row 2, column 3)
01
Given the matrix \(B = \begin{bmatrix} 4 & -2 & 9 \\ 1 & 0 & -5 \\ 6 & 3 & 8 \end{bmatrix}\),
what is the element \(b_{32}\)?
02
Matrix \(P\) has 12 elements arranged in 3 rows. Which statement is correct?
02
Matrix Addition & Scalar Multiplication
⚡ Quick Memory Point
SAME SIZE → ADD. SCALAR → MULTIPLY EVERY entry.
Addition rule: Must be same dimension. Add element-by-element.
Scalar rule: k·A → multiply every single entry by k
Trick: \(-A\) = scalar \((-1) \cdot A\) → flip all signs
Scalar rule: k·A → multiply every single entry by k
Trick: \(-A\) = scalar \((-1) \cdot A\) → flip all signs
📐 Worked Example
If \(A = \begin{bmatrix} 2 & 5 \\ -1 & 3 \end{bmatrix}\) and \(B = \begin{bmatrix} 4 & -2 \\ 0 & 6 \end{bmatrix}\),
\(A + B = \begin{bmatrix} 2+4 & 5+(-2) \\ -1+0 & 3+6 \end{bmatrix} = \begin{bmatrix} 6 & 3 \\ -1 & 9 \end{bmatrix}\)
\(3A = \begin{bmatrix} 6 & 15 \\ -3 & 9 \end{bmatrix}\) ← every entry × 3
\(A + B = \begin{bmatrix} 2+4 & 5+(-2) \\ -1+0 & 3+6 \end{bmatrix} = \begin{bmatrix} 6 & 3 \\ -1 & 9 \end{bmatrix}\)
\(3A = \begin{bmatrix} 6 & 15 \\ -3 & 9 \end{bmatrix}\) ← every entry × 3
03
If \(A = \begin{bmatrix} 3 & -1 \\ 2 & 5 \end{bmatrix}\) and \(B = \begin{bmatrix} -3 & 4 \\ 1 & -2 \end{bmatrix}\),
find \(2A - B\).
04
Which of the following is always true?
03
Matrix Multiplication
⚡ Quick Memory Point
"INNER DIMENSIONS must match — OUTER gives size!"
A (m×n) · B (n×p) = C (m×p)
Inner: n = n ✓ → can multiply
Outer: m × p = size of result
WARNING: AB ≠ BA in general (not commutative!)
Element trick: Row of A · Column of B = one number
Inner: n = n ✓ → can multiply
Outer: m × p = size of result
WARNING: AB ≠ BA in general (not commutative!)
Element trick: Row of A · Column of B = one number
📐 Worked Example
\(A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\), \(B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}\)
\(AB = \begin{bmatrix} (1)(5)+(2)(7) & (1)(6)+(2)(8) \\ (3)(5)+(4)(7) & (3)(6)+(4)(8) \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}\)
Check: (2×2)·(2×2) → inner 2=2 ✓ → result is 2×2
\(AB = \begin{bmatrix} (1)(5)+(2)(7) & (1)(6)+(2)(8) \\ (3)(5)+(4)(7) & (3)(6)+(4)(8) \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}\)
Check: (2×2)·(2×2) → inner 2=2 ✓ → result is 2×2
05
Matrix \(A\) is \(2 \times 3\) and matrix \(B\) is \(3 \times 4\). Which products are defined?
06
Compute \(AB\) where \(A = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}\) and \(B = \begin{bmatrix} 4 \\ -2 \end{bmatrix}\).
07
Given \(A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\) and \(B = \begin{bmatrix} 3 & 0 \\ 1 & 2 \end{bmatrix}\),
find the (1,1) entry of \(BA - AB\).
04
Determinant & Inverse
⚡ Quick Memory Point
"AD minus BC" — the det(2×2) chant
For \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\): det = ad − bc
Inverse formula: \(A^{-1} = \dfrac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\)
Memory: SWAP a↔d, FLIP signs of b & c, then divide by det
det = 0 → NO inverse! (singular matrix)
Inverse formula: \(A^{-1} = \dfrac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\)
Memory: SWAP a↔d, FLIP signs of b & c, then divide by det
det = 0 → NO inverse! (singular matrix)
📐 Worked Example
Find \(A^{-1}\) for \(A = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}\)
Step 1: \(\det(A) = (3)(2) - (1)(5) = 6 - 5 = 1\)
Step 2: Swap 3↔2, flip signs: \(\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}\)
Step 3: \(A^{-1} = \frac{1}{1}\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}\)
✅ Check: \(AA^{-1} = I\)
Step 1: \(\det(A) = (3)(2) - (1)(5) = 6 - 5 = 1\)
Step 2: Swap 3↔2, flip signs: \(\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}\)
Step 3: \(A^{-1} = \frac{1}{1}\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}\)
✅ Check: \(AA^{-1} = I\)
08
Find \(\det\begin{bmatrix} 5 & -2 \\ 3 & 4 \end{bmatrix}\).
09
Find \(A^{-1}\) for \(A = \begin{bmatrix} 4 & 7 \\ 2 & 5 \end{bmatrix}\).
10
For what value of \(k\) does the matrix \(\begin{bmatrix} k & 4 \\ 3 & 6 \end{bmatrix}\) have no inverse?
05
Systems of Equations & Matrix Methods
⚡ Quick Memory Point
AX = B → X = A⁻¹B (multiply LEFT by inverse!)
System: ax + by = e
cx + dy = f
Matrix form: \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}e\\f\end{bmatrix}\)
Cramer's Rule: \(x = \dfrac{\det(A_x)}{\det(A)}\), \(y = \dfrac{\det(A_y)}{\det(A)}\)
cx + dy = f
Matrix form: \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}e\\f\end{bmatrix}\)
Cramer's Rule: \(x = \dfrac{\det(A_x)}{\det(A)}\), \(y = \dfrac{\det(A_y)}{\det(A)}\)
📐 Worked Example
Solve: \(2x + y = 5\), \(x - y = 1\) using matrices
\(A = \begin{bmatrix}2&1\\1&-1\end{bmatrix}\), \(B = \begin{bmatrix}5\\1\end{bmatrix}\)
\(\det(A) = -2-1 = -3\)
\(A^{-1} = \frac{1}{-3}\begin{bmatrix}-1&-1\\-1&2\end{bmatrix}\)
\(X = A^{-1}B = \frac{1}{-3}\begin{bmatrix}-6\\3\end{bmatrix} = \begin{bmatrix}2\\-1\end{bmatrix}\)
So \(x = 2, y = -1\)
\(A = \begin{bmatrix}2&1\\1&-1\end{bmatrix}\), \(B = \begin{bmatrix}5\\1\end{bmatrix}\)
\(\det(A) = -2-1 = -3\)
\(A^{-1} = \frac{1}{-3}\begin{bmatrix}-1&-1\\-1&2\end{bmatrix}\)
\(X = A^{-1}B = \frac{1}{-3}\begin{bmatrix}-6\\3\end{bmatrix} = \begin{bmatrix}2\\-1\end{bmatrix}\)
So \(x = 2, y = -1\)
11
Write the system \(3x - 2y = 7\) and \(x + 4y = -1\) in matrix form \(AX = B\). Which is correct?
12
Using \(A^{-1}\), solve the system: \(5x + 3y = 11\) and \(2x + y = 4\).
13
Using Cramer's Rule on \(2x + y = 8\) and \(x - 3y = -1\), find \(x\).
06
Advanced Topics & Word Problems
⚡ Quick Memory Point
Transpose: Rows ↔ Columns. Identity: Like multiplying by 1.
Transpose \(A^T\): Flip matrix over its diagonal (row i becomes col i)
Identity \(I\): 1s on diagonal, 0s elsewhere → \(AI = IA = A\)
Property: \((AB)^T = B^T A^T\) (order reverses!)
Property: \((A^{-1})^T = (A^T)^{-1}\)
Identity \(I\): 1s on diagonal, 0s elsewhere → \(AI = IA = A\)
Property: \((AB)^T = B^T A^T\) (order reverses!)
Property: \((A^{-1})^T = (A^T)^{-1}\)
14
If \(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\), which is \(A^T\)?
15
If \(AI = A\) and \(IA = A\), what must be true about \(I\)?
16
🏪 Real-World Scenario
A café sells two drinks: coffee at $4 each and tea at $2.50 each.
On Monday, total sales were $47.50, and 15 drinks were sold in total.
Let \(x\) = number of coffees, \(y\) = number of teas.
Set up and solve using matrix methods.
How many coffees were sold?
17
💰 Real-World Scenario
A student invests a total of $10,000 into two accounts.
Account A earns 4% annual interest and Account B earns 6% annual interest.
After one year, the total interest earned is $520.
Using matrix equation \(AX = B\), how much was invested in Account A?
18
🥗 Real-World Scenario
A nutritionist creates meal plans using two foods. Food X has 3g protein and 2g fat per serving. Food Y has 5g protein and 1g fat per serving. A meal must contain exactly 19g protein and 8g fat.
Using matrix methods, how many servings of Food X are needed?
19
If \(\det(A) = 3\) and \(\det(B) = -2\) (both \(2\times 2\)), what is \(\det(2AB)\)?
20
🏭 Real-World Scenario — Final Boss
A factory makes two products, P and Q.
Each unit of P requires 2 hours of machining and 1 hour of assembly.
Each unit of Q requires 1 hour of machining and 3 hours of assembly.
In a week, 20 hours of machining and 27 hours of assembly are available.
Set up the system and use matrices to find how many units of each product to make.
How many units of Q should be produced?
Quiz Complete! 🎓
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Keep practicing — matrices unlock the language of the universe.