Question

# Detenninants of Sums Give an example of square matrices A and B for which |A+B|neq |A|+|B|

Matrices
Detenninants of Sums Give an example of square matrices A and B for which $$|A+B|\neq |A|+|B|$$

## Expert Answers (1)

2021-02-04

Step 1
Let matrix A and B as:
$$A=\begin{bmatrix}2 & 3 \\5& 1 \end{bmatrix}$$
$$B=\begin{bmatrix}1 & 2 \\0 & 1 \end{bmatrix}$$ Step 2 The summation of matrices:
$$A+B=\begin{bmatrix}2 & 3 \\5& 1 \end{bmatrix}+ \begin{bmatrix}1 & 2 \\0 & 1 \end{bmatrix}$$
$$A+B=\begin{bmatrix}3 & 5 \\5 & 2 \end{bmatrix}$$
Step 3
Determinant of summation of matrices:
$$|A+B|=3(2)-5(5)=6-25=-19$$
Step 4
Determinant of matrix A and matrix B
$$|A|=\begin{bmatrix}2 & 3 \\5& 1 \end{bmatrix}=2(1)-3(5)=2-15=-13$$
$$|B|=\begin{bmatrix}1 & 2 \\0 & 1 \end{bmatrix}=1(1)-2(0)=1-0=1$$

Step 5
Hence, it is clear that:
$$|A+B|\neq |A|+|B|$$