Dolly Robinson

2020-12-17

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.
(a) Addition of matrices is not commutative.
(b) The transpose of the sum of matrices is equal to the sum of the transposes of the matrices.

a) False.
The addition of Matrices is Commutative.
For example:
Consider two matrices, of order
$A=\left[\left[1,3\right],\left[2,4\right]\right],B=\left[\left[0,1\right],\left[-1,4\right]\right]$
Matrices A and B are commutative under addition, if, A+B=B+A
$⇒\left[\left[1,3\right],\left[2,4\right]\right]+\left[\left[0,1\right],\left[-1,4\right]\right]=\left[\left[0,1\right],\left[-1,4\right]\right]+\left[\left[1,3\right],\left[2,4\right]\right]$
$⇒\left[\left[1,4\right],\left[1,8\right]\right]=\left[\left[1,4\right],\left[1,8\right]\right]$
Hence Commutative.
b) True.
Yes, The transpose of the sum of matrices is equal to the sum of the transposes of the matrices.
Consider two matrices, of order $2×2$
$A=\left[\left[1,3\right],\left[2,4\right]\right],B=\left[\left[0,1\right],\left[-1,4\right]\right]$
$A+B=\left[\left[1,4\right],\left[1,8\right]\right]$
$⇒\left(A+B{\right)}^{\prime }=\left[\left[1,1\right],\left[4,8\right]\right]$ (1)
${A}^{\prime }=\left[\left[1,2\right],\left[3,4\right]\right]$
${B}^{\prime }=\left[\left[0,-1\right],\left[1,4\right]\right]$
${A}^{\prime }+{B}^{\prime }=\left[\left[1,2\right],\left[3,4\right]\right]+\left[\left[0,-1\right],\left[1,4\right]\right]$
$⇒{A}^{\prime }+{B}^{\prime }=\left[\left[1,1\right],\left[4,8\right]\right]$ (2)
Drom (1) and (2)
(A+B)'=A'+B'
Hence proved

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