Answer to Question #118745 in Linear Algebra for Gly

Question

Given the matrices (Each matrices are inside a [ ] )
3 0 1 5 2 0 1 3
A= -1 2 , B= 4 -1 , C= 1 4 2 , D= -1 0 1 , E= -1 1 2
1 1 0 2 3 1 5 3 2 4 4 1 3

Compute if possible
1. -4C^2
2. (E-D)^T
3. A(BC)
4. (B^T B)C
5. 3B-AB

Answer 1

$A=\begin{pmatrix} 3 & 0 \\ -1 & 2\\ 1&1 \end{pmatrix}, B=\begin{pmatrix} 1 & 5 \\ 4 & -1\\ 0&2 \end{pmatrix}, \\ C=\begin{pmatrix} 2 & 0&1 \\ 1 & 4&2\\ 3&1&5 \end{pmatrix}, D=\begin{pmatrix} -1 &0&1 \\ 3 & 2&4 \end{pmatrix}, \\ E=\begin{pmatrix} -1 &1&2 \\ 4 & 1&3 \end{pmatrix}$

$-4C^2=-4\begin{pmatrix} 2 & 0&1 \\ 1 & 4&2\\ 3&1&5 \end{pmatrix}\cdot\begin{pmatrix} 2 & 0&1 \\ 1 & 4&2\\ 3&1&5 \end{pmatrix}=\\ =-4\begin{pmatrix} 4+0+3 & 0+0+1&2+0+5 \\ 2+4+6 & 0+16+2&1+8+10\\ 6+1+15&0+4+5&3+2+25 \end{pmatrix}=\\ =-4\begin{pmatrix} 7 & 1&7 \\ 12 & 18&19\\ 22&9&30 \end{pmatrix}=\\ =\begin{pmatrix} - 28 & -4&-28 \\ -48 & -72&-76\\ -88&-36&-120 \end{pmatrix}$

2.

$(E-D)^T=\\=\left(\begin{pmatrix} -1 &1&2 \\ 4 & 1&3 \end{pmatrix}-\begin{pmatrix} -1 &0&1 \\ 3 & 2&4 \end{pmatrix}\right)^T=\\ =\left(\begin{pmatrix} 0 &1&1 \\ 1 & -1&-1 \end{pmatrix}\right)^T=\\ =\begin{pmatrix} 0 & 1 \\ 1& -1\\ 1&-1 \end{pmatrix}$

3.

$A(BC)$

$B:3\times2, C:3\times3\\$

The product $BC$ does not exist.

4.

$(B^TB)C$

$B^T=\begin{pmatrix} 1 & 4&0 \\ 5 & -1&2 \end{pmatrix},B=\begin{pmatrix} 1 & 5 \\ 4 & -1\\ 0&2 \end{pmatrix}\\ B^T\cdot B=\begin{pmatrix} 1 & 4&0 \\ 5 & -1&2 \end{pmatrix}\cdot\begin{pmatrix} 1 & 5 \\ 4 & -1\\ 0&2 \end{pmatrix}=\\ = \begin{pmatrix} 1+16+0 & 5-4+0 \\ 5-4+0 & 25+1+4 \end{pmatrix}=\begin{pmatrix} 17 & 1 \\ 1 & 30 \end{pmatrix}$

$B^T\cdot B:2\times2, C:3\times3$

The product $(B^TB)C$ does not exist.

5.

$3B-AB\\ 3B=3\begin{pmatrix} 1 & 5 \\ 4 & -1\\ 0&2 \end{pmatrix}=\begin{pmatrix} 3 & 15 \\ 12 & -3\\ 0&6 \end{pmatrix}$

$A:3\times2, B:3\times2$

The product $AB$ does not exist.

Then 3B-AB does not exist.