Adding and Subtracting Matrices with Like Dimensions

Chapter 24

Concept Explanation:

Adding and subtracting matrices with like dimensions means that both matrices must have the same number of rows and columns. Each element in the resulting matrix is calculated by performing the addition or subtraction of the corresponding elements from the two matrices.

Common Mistakes:

Dimension Mismatch: Ensure both matrices have the same dimensions before performing operations.
Element Misalignment: Always check that you are adding or subtracting the correct corresponding elements.

Helpful Tips:

Organize Your Work: Write out both matrices clearly and align them to avoid mistakes.
Practice with Different Sizes: Work with matrices of varying sizes to build confidence.

Hard Questions:

Q1: Add $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$ .

Step-by-Step Solution:

Add the matrices element-wise:

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

Answer:

The resulting matrix is $\begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$ .

Q2: Subtract $C = \begin{bmatrix} 9 & 4 \ 6 & 2 \end{bmatrix}$ from $D = \begin{bmatrix} 3 & 7 \ 1 & 5 \end{bmatrix}$ .

Step-by-Step Solution:

Subtract the matrices element-wise:

$D - C = \begin{bmatrix} 3 - 9 & 7 - 4 \ 1 - 6 & 5 - 2 \end{bmatrix} = \begin{bmatrix} -6 & 3 \ -5 & 3 \end{bmatrix}$

Answer:

The resulting matrix is $\begin{bmatrix} -6 & 3 \ -5 & 3 \end{bmatrix}$ .

Q3: Add $E = \begin{bmatrix} 2 & 3 \ 1 & 0 \end{bmatrix}$ and $F = \begin{bmatrix} 4 & -1 \ -3 & 5 \end{bmatrix}$ .

Step-by-Step Solution:

Add the matrices element-wise:

$E + F = \begin{bmatrix} 2 + 4 & 3 - 1 \ 1 - 3 & 0 + 5 \end{bmatrix} = \begin{bmatrix} 6 & 2 \ -2 & 5 \end{bmatrix}$

Answer:

The resulting matrix is $\begin{bmatrix} 6 & 2 \ -2 & 5 \end{bmatrix}$ .

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