Adding and Subtracting Matrices

Chapter 16

Adding and Subtracting Matrices Concept Explanation

Adding and subtracting matrices involves element-wise operations. This means that to add or subtract two matrices, they must be of the same dimensions. Each element in one matrix is added or subtracted from the corresponding element in the other matrix.

Adding matrices:

$C_{ij} = A_{ij} + B_{ij}$

Subtracting matrices:

$C_{ij} = A_{ij} - B_{ij}$

Diagram:

Here’s an example of adding two matrices:

$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$

$A + B = \begin{bmatrix} 1 + 5 & 2 + 6 \ 3 + 7 & 4 + 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$

Common Mistakes:

Dimension Mismatch: Always check if the matrices have the same dimensions before attempting to add or subtract them.
Element Misalignment: Ensure you are adding or subtracting the correct corresponding elements.

Hard Questions:

Q1: Add the matrices $A = \begin{bmatrix} 3 & 4 \ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & -1 \ 0 & 6 \end{bmatrix}$ .

Step-by-Step Solution:

Add the matrices element-wise:

$C = A + B = \begin{bmatrix} 3 + 5 & 4 + (-1) \ 1 + 0 & 2 + 6 \end{bmatrix} = \begin{bmatrix} 8 & 3 \ 1 & 8 \end{bmatrix}$

Answer:

The result is $C = \begin{bmatrix} 8 & 3 \ 1 & 8 \end{bmatrix}$ .

Q2: Subtract the matrix $B = \begin{bmatrix} 4 & 3 \ 2 & 1 \end{bmatrix}$ from $A = \begin{bmatrix} 7 & 5 \ 3 & 4 \end{bmatrix}$ .

Step-by-Step Solution:

Subtract the matrices element-wise:

$C = A - B = \begin{bmatrix} 7 - 4 & 5 - 3 \ 3 - 2 & 4 - 1 \end{bmatrix} = \begin{bmatrix} 3 & 2 \ 1 & 3 \end{bmatrix}$

Answer:

The result is $C = \begin{bmatrix} 3 & 2 \ 1 & 3 \end{bmatrix}$ .

Q3: Add $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 7 & 8 & 9 \ 10 & 11 & 12 \end{bmatrix}$ .

Step-by-Step Solution:

Add the matrices element-wise:

$C = A + B = \begin{bmatrix} 1 + 7 & 2 + 8 & 3 + 9 \ 4 + 10 & 5 + 11 & 6 + 12 \end{bmatrix} = \begin{bmatrix} 8 & 10 & 12 \ 14 & 16 & 18 \end{bmatrix}$

Answer:

The result is $C = \begin{bmatrix} 8 & 10 & 12 \ 14 & 16 & 18 \end{bmatrix}$ .

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