Question

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

Matrices
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asked 2021-02-03
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:
\(|A+B|\neq |A|+|B|\)

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