Question

A=[3 0 2]

[4 -6 3]

[-2 1 8]

B=[-5 1 1]

[0 3 0]

[7 6 2]

C=[1 1 1]

[2 3 - 1]

[3 - 5 - 7]

Verify the following expressions(where possible and give reasons)

(i) A+(B+C) =(A+B) +C and A(BC) =(AB) C

(ii) (a-b) C=aC - bC and a(B - C) =aB - aC

(iii) (A^T) ^T=A and (A - B) ^T=A^T - B^T

Answer 1

A = $\begin{bmatrix} 3 & 0 & 2 \\ 4 & -6 & 3\\ -2 & 1 & 8\\ \end{bmatrix}$

B = $\begin{bmatrix} -5 & 1 &1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \\ \end{bmatrix}$

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

(i) A+(B+C) =(A+B) +C and A(BC) =(AB) C

a) A+(B+C) =(A+B) +C

LHS:

= A+(B+C)

=$\begin{bmatrix} 3 & 0 & 2 \\ 4 & -6 & 3\\ -2 & 1 & 8\\ \end{bmatrix}$ + $\begin{bmatrix} -4 & 2 &2 \\ 2 & 6 & -1 \\ 10 & 1 & -5 \\ \end{bmatrix}$

= $\begin{bmatrix} -1 & 2 &4\\ 6 & 0 &2\\ 8&2&3 \end{bmatrix}$

RHS:

(A+B) +C

= $\begin{bmatrix} -2&1&3\\ 4&-3&3\\ 5&7&10 \end{bmatrix}$ + $\begin{bmatrix} 1 & 1 &1\\ 2 &3&-1\\ 3&-5&-7 \end{bmatrix}$

= $\begin{bmatrix} -1 & 2 &4\\ 6 & 0 &2\\ 8&2&3 \end{bmatrix}$

LHS = RHS

b) A(BC) =(AB) C

LHS:

= $\begin{bmatrix} 3 & 0 & 2 \\ 4 & -6 & 3\\ -2 & 1 & 8\\ \end{bmatrix}$ * $\begin{bmatrix} 0&-7&-13\\ 6&9&-3\\ 25&15&-13\\ \end{bmatrix}$

= $\begin{bmatrix} 50&9&-65\\ 39&-37&-73\\ 206&143&-81\\ \end{bmatrix}$

RHS:

= $\begin{bmatrix} -1 & 15&7 \\ 1 & 4&10\\ 66&49&14\\ \end{bmatrix}$ * $\begin{bmatrix} 1 & 1 &1\\ 2 &3&-1\\ 3&-5&-7 \end{bmatrix}$

= $\begin{bmatrix} 50&9&-65\\ 39&-37&-73\\ 206&143&-81\\ \end{bmatrix}$

LHS = RHS

(ii) (a-b) C=aC - bC and a(B - C) =aB - aC

a) (a-b) C=aC - bC

LHS:

= (a - b) * $\begin{bmatrix} 1 & 1 &1\\ 2 &3&-1\\ 3&-5&-7 \end{bmatrix}$

= $\begin{bmatrix} a-b & a-b &a-b\\ 2(a - b) &3(a - b)&-1(a - b)\\ 3(a - b)&-5(a - b)&-7(a - b)\\ \end{bmatrix}$

RHS:

= a * $\begin{bmatrix} 1 & 1 &1\\ 2 &3&-1\\ 3&-5&-7 \end{bmatrix}$ - b * $\begin{bmatrix} 1 & 1 &1\\ 2 &3&-1\\ 3&-5&-7 \end{bmatrix}$

= $\begin{bmatrix} a & a &a\\ 2a &3a&-a\\ 3a&-5a&-7a \end{bmatrix}$ - $\begin{bmatrix} b & b &b\\ 2b &3b&-b\\ 3b&-5b&-7b \end{bmatrix}$

= $\begin{bmatrix} a-b & a-b &a-b\\ 2(a - b) &3(a - b)&-1(a - b)\\ 3(a - b)&-5(a - b)&-7(a - b)\\ \end{bmatrix}$

LHS = RHS

b) a(B - C) =aB - aC

LHS:

= a * $\begin{bmatrix} -6&0&0\\ -2&0&1\\ 4&11&9 \end{bmatrix}$

= $\begin{bmatrix} -6a&0&0\\ -2a&0&a\\ 4a&11a&9a \end{bmatrix}$

RHS:

= aB - aC

= a * $\begin{bmatrix} -5 & 1 &1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \\ \end{bmatrix}$ - a * $\begin{bmatrix} 1 & 1 &1\\ 2 &3&-1\\ 3&-5&-7 \end{bmatrix}$

= $\begin{bmatrix} -6a&0&0\\ -2a&0&a\\ 4a&11a&9a \end{bmatrix}$

LHS = RHS

(iii) (A^T) ^T=A and (A - B) ^T=A^T - B^T

a) (A^T) ^T=A

LHS:

= ( A^T ) ^T

= $(\begin{bmatrix} 3 & 4 & -2 \\ 4 & -6 & 1\\ -2 & 1 & 8\\ \end{bmatrix} )^T$

= $\begin{bmatrix} 3 & 4 & -2 \\ 4 & -6 & 1\\ -2 & 1 & 8\\ \end{bmatrix}$ = A

LHS = RHS

(b) (A - B) ^T=A^T - B^T

LHS:

= $(\begin{bmatrix} 8 & -1 & 1 \\ 4 & -9 & 3\\ -9 & -5 & 6\\ \end{bmatrix} )^T$

= $\begin{bmatrix} 8 & 4 & -9 \\ -1 & -9 & -5\\ 1 & 3 & 6\\ \end{bmatrix}$

RHS:

= A^T - B^T

= $(\begin{bmatrix} 3 & 0 & 2 \\ 4 & -6 & 3\\ -2 & 1 & 8\\ \end{bmatrix} ) ^T$ - $(\begin{bmatrix} -5 & 1 &1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \\ \end{bmatrix})^T$

= $\begin{bmatrix} 8 & 4 & -9 \\ -1 & -9 & -5\\ 1 & 3 & 6\\ \end{bmatrix}$

LHS = RHS