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

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

2Problem 2medium

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

3Problem 3hard

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

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