Matrix Operations

Adding, subtracting, and multiplying matrices

Matrix Operations

What is a Matrix?

A matrix is a rectangular array of numbers.

Example: $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

Dimensions: rows × columns (this is a 2×2 matrix)

Adding Matrices

Add corresponding entries. Matrices must have same dimensions.

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} a+e & b+f \\ c+g & d+h \end{bmatrix}$

Subtracting Matrices

Subtract corresponding entries.

$\begin{bmatrix} 5 & 3 \\ 2 & 1 \end{bmatrix} - \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$

Scalar Multiplication

Multiply every entry by the scalar:

$3\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix}$

Matrix Multiplication

Not commutative! $AB \neq BA$ in general

For $A_{m \times n}$ and $B_{n \times p}$ :

Result is $m \times p$ matrix
Inner dimensions must match!

Entry formula: $(AB)_{ij} =$ (row $i$ of $A$ ) · (column $j$ of $B$ )

Example: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

📚 Practice Problems

1Problem 1easy

❓ Question:

Add the matrices: [2 3] + [1 4] [5 1] [2 0]

💡 Show Solution

Step 1: Add corresponding elements: [2+1 3+4] = [3 7] [5+2 1+0] [7 1]

Step 2: Note that matrices must have the same dimensions to be added

Answer: [3 7] [7 1]

2Problem 2easy

❓ Question:

Add: $\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix} + \begin{bmatrix} 4 & 1 \\ 2 & 6 \end{bmatrix}$

💡 Show Solution

Add corresponding entries:

$\begin{bmatrix} 2+4 & 5+1 \\ 1+2 & 3+6 \end{bmatrix} = \begin{bmatrix} 6 & 6 \\ 3 & 9 \end{bmatrix}$

Answer: $\begin{bmatrix} 6 & 6 \\ 3 & 9 \end{bmatrix}$

3Problem 3easy

❓ Question:

Multiply the matrix by the scalar: 3[2 -1] [4 5]

💡 Show Solution

Step 1: Multiply each element by 3: 3[2 -1] = [3·2 3·(-1)] [4 5] [3·4 3·5]

Step 2: Calculate: = [6 -3] [12 15]

Answer: [6 -3] [12 15]

4Problem 4medium

❓ Question:

Multiply: $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \begin{bmatrix} 5 \\ 6 \end{bmatrix}$

💡 Show Solution

This is a 2×2 matrix times a 2×1 matrix. Result will be 2×1.

First entry: $2(5) + 3(6) = 10 + 18 = 28$

Second entry: $1(5) + 4(6) = 5 + 24 = 29$

Answer: $\begin{bmatrix} 28 \\ 29 \end{bmatrix}$

5Problem 5medium

❓ Question:

Multiply: [1 2][5] [3 4][6]

💡 Show Solution

Step 1: Check dimensions: First matrix: 2×2 Second matrix: 2×1 Result will be: 2×1

Step 2: Calculate first element (row 1 × column): 1(5) + 2(6) = 5 + 12 = 17

Step 3: Calculate second element (row 2 × column): 3(5) + 4(6) = 15 + 24 = 39

Step 4: Write result: [17] [39]

Answer: [17] [39]

6Problem 6medium

❓ Question:

Multiply: [1 2][5 6] [3 4][7 8]

💡 Show Solution

Step 1: Check dimensions: Both are 2×2, result will be 2×2

Step 2: Calculate element (1,1): Row 1 × Column 1: 1(5) + 2(7) = 5 + 14 = 19

Step 3: Calculate element (1,2): Row 1 × Column 2: 1(6) + 2(8) = 6 + 16 = 22

Step 4: Calculate element (2,1): Row 2 × Column 1: 3(5) + 4(7) = 15 + 28 = 43

Step 5: Calculate element (2,2): Row 2 × Column 2: 3(6) + 4(8) = 18 + 32 = 50

Step 6: Write result: [19 22] [43 50]

Answer: [19 22] [43 50]

7Problem 7hard

❓ Question:

Multiply: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$

💡 Show Solution

Calculate each entry of the 2×2 result:

Entry (1,1): $1(2) + 2(1) = 2 + 2 = 4$

Entry (1,2): $1(0) + 2(3) = 0 + 6 = 6$

Entry (2,1): $3(2) + 4(1) = 6 + 4 = 10$

Entry (2,2): $3(0) + 4(3) = 0 + 12 = 12$

Answer: $\begin{bmatrix} 4 & 6 \\ 10 & 12 \end{bmatrix}$

8Problem 8hard

❓ Question:

Given A = [1 2] and B = [5 6], find 2A - 3B. [3 4] [7 8]

💡 Show Solution

Step 1: Calculate 2A: 2A = 2[1 2] = [2 4] [3 4] [6 8]

Step 2: Calculate 3B: 3B = 3[5 6] = [15 18] [7 8] [21 24]

Step 3: Subtract 3B from 2A: 2A - 3B = [2 4] - [15 18] [6 8] [21 24]

Step 4: Subtract corresponding elements: = [2-15 4-18] [6-21 8-24]

Step 5: Simplify: = [-13 -14] [-15 -16]

Answer: [-13 -14] [-15 -16]

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