Question 1: State True or False. If False, please state the reason.

1. If $A$ and $B$ are two matrices of order $3 \times 2$ and $2 \times 3$ respectively; then their sum $A + B$ is possible.
2. The Matrices $A_{2 \times 3}$ and  $A_{2 \times 3}$ are conformable for subtraction.
3. Transpose of a $2 \times 1$ matrix is a $2\times 1$ matrix.
4. Transpose of a square matrix is a square matrix.
5. A column matrix has many columns and only one row.

1. False: Two matrices can be added together if they are of the same order. Here $A$ is of the Order $3 \times 2$ while $B$ is of the order $2 \times 3$. Hence they cannot be added.
2. True: Two matrices can be subtracted together if they are of the same order. Here both latex A  &s=0$and latex B &s=0$ are of the same order.
3. False: The transpose of a matrix is obtained by interchanging rows with columns. Hence the Transpose of a $2 \times 1$ matrix is a $1 \times 2$ matrix.
4. True: Yes Transpose of a square matrix is a square matrix. Here the number of rows is equal to the number of columns. Hence even on transposing, the matrix would remain as a square matrix.
5. False: A Column matrix has one column and many rows.

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Question 2: Given $\begin{bmatrix} x & y+2 \\ 3 & z-1 \end{bmatrix} = \begin{bmatrix} 3 & 1 \\ 3 & 2 \end{bmatrix}$, find $x, \ y \ and \ z$.

$\begin{bmatrix} x & y+2 \\ 3 & z-1 \end{bmatrix} = \begin{bmatrix} 3 & 1 \\ 3 & 2 \end{bmatrix}$

$\Rightarrow x = 3$

$y+2 = 1 \ \Rightarrow y = -1$

also  $z-1 = 2 \Rightarrow z = 3$

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Question 3: Solve for $a, \ b \ and \ c$ if;

1. $\begin{bmatrix} -4 & a+5 \\ 3 & 2 \end{bmatrix} = \begin{bmatrix} b+4 & 2 \\ 3 & c-1 \end{bmatrix}$
2. $\begin{bmatrix} a & a-b \\ b+c & 0 \end{bmatrix} = \begin{bmatrix} 3 & -1 \\ 2 & 0 \end{bmatrix}$

1)     Given $\begin{bmatrix} -4 & a+5 \\ 3 & 2 \end{bmatrix} = \begin{bmatrix} b+4 & 2 \\ 3 & c-1 \end{bmatrix}$; Therefore

$-4 = b+ 4 \Rightarrow b = -8$

$a+5 = 2 \Rightarrow a = -3$

$2 = c-1 \Rightarrow c = 3$

2)    $\begin{bmatrix} a & a-b \\ b+c & 0 \end{bmatrix} = \begin{bmatrix} 3 & -1 \\ 2 & 0 \end{bmatrix}$

$a = 3$

$a-b=-1 \Rightarrow b = a-1 = 3-1 = 2$

$b+c = 2 \Rightarrow c = 2-b = 2-2 = 0$

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Question 4: If  $A = \begin{bmatrix} 8 & -3 \end{bmatrix}$ and  $B = \begin{bmatrix} 4 & -5 \end{bmatrix}$  find

1. $A+B$
2. $B-A$

1)    $A+B$

$= \begin{bmatrix} 8 & -3 \end{bmatrix} + \begin{bmatrix} 4 & -5 \end{bmatrix}$

$=\begin{bmatrix} 8+4 & -3-5 \end{bmatrix} = \begin{bmatrix} 12 & -8 \end{bmatrix}$

2)    $B-A$

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

$=\begin{bmatrix} 4-8 & -5-(-3) \end{bmatrix} = \begin{bmatrix} -4 & -2 \end{bmatrix}$

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Question 5: If $A = \begin{bmatrix} 2 \\ 5 \end{bmatrix}, \ B=\begin{bmatrix} 1 \\ 4 \end{bmatrix} \ and \ C=\begin{bmatrix} 6 \\ -2 \end{bmatrix}$ find:

1. $B+C$
2. $A-C$
3. $A+B-C$
4. $A-B+C$

1)    $B+C$

$= \begin{bmatrix} 1 \\ 4 \end{bmatrix} + \begin{bmatrix} 6 \\ -2 \end{bmatrix}$

$= \begin{bmatrix} 1+6 \\ 4-2 \end{bmatrix}$

$= \begin{bmatrix} 7 \\ 2 \end{bmatrix}$

2)    $A-C$

$= \begin{bmatrix} 2 \\ 5 \end{bmatrix} - \begin{bmatrix} 6 \\ -2 \end{bmatrix}$

$= \begin{bmatrix} 2-6 \\ 5-(-2) \end{bmatrix}$

$= \begin{bmatrix} -4 \\ 7 \end{bmatrix}$

3)    $A+B-C$

$= \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} 1 \\ 4 \end{bmatrix} - \begin{bmatrix} 6 \\ -2 \end{bmatrix}$

$= \begin{bmatrix} 2+1-6 \\ 5+4-(-2) \end{bmatrix}$

$= \begin{bmatrix} -3 \\ 11 \end{bmatrix}$

4)    $A-B+C$

$= \begin{bmatrix} 2 \\ 5 \end{bmatrix} - \begin{bmatrix} 1 \\ 4 \end{bmatrix} + \begin{bmatrix} 6 \\ -2 \end{bmatrix}$

$= \begin{bmatrix} 2-1+6 \\ 5-4+(-2) \end{bmatrix}$

$= \begin{bmatrix} 7 \\ -1 \end{bmatrix}$

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Question 6: Wherever possible, write each of the following in a single matrix:

1. $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} -1 & -2 \\ 1 & -7 \end{bmatrix}$
2. $\begin{bmatrix} 2 &3 & 4 \\ 5 & 6 & 7 \end{bmatrix} - \begin{bmatrix} 0 &2 & 3 \\ 6 & -1 & 0 \end{bmatrix}$
3. $\begin{bmatrix} 0 & 1 & 2 \\ 4 & 6 & 7 \end{bmatrix} + \begin{bmatrix} 3 & 4 \\ 6 & 8 \end{bmatrix}$

1) $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} -1 & -2 \\ 1 & -7 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 4 & -3 \end{bmatrix}$

2) $\begin{bmatrix} 2 &3 & 4 \\ 5 & 6 & 7 \end{bmatrix} - \begin{bmatrix} 0 &2 & 3 \\ 6 & -1 & 0 \end{bmatrix} = \begin{bmatrix} 2 &5 & 7 \\ 11 & 5 & 7 \end{bmatrix}$

3) Adding this is is not possible as the order of the metrices are not the same.

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Question 7: Find $x$ and $y$ from the following equations:

1. $\begin{bmatrix} 5 & 2 \\ -1 & y-1 \end{bmatrix} - \begin{bmatrix} 1 & x-1 \\ 2 & -3 \end{bmatrix} = \begin{bmatrix} 4 & 7 \\ -3 & 2 \end{bmatrix}$
2. $\begin{bmatrix} -8 & x \end{bmatrix} + \begin{bmatrix} y & -2 \end{bmatrix} = \begin{bmatrix} -3 & 2 \end{bmatrix}$

1)     $\begin{bmatrix} 5 & 2 \\ -1 & y-1 \end{bmatrix} - \begin{bmatrix} 1 & x-1 \\ 2 & -3 \end{bmatrix} = \begin{bmatrix} 4 & 7 \\ -3 & 2 \end{bmatrix}$

$\begin{bmatrix} 5-1 & 2-(x-1) \\ (-1-2) & y-1-(-3) \end{bmatrix} =\begin{bmatrix} 4 & 7 \\ -3 & 2 \end{bmatrix}$

$= \begin{bmatrix} 4 & 3-x \\ -3 & y+2 \end{bmatrix} =\begin{bmatrix} 4 & 7 \\ -3 & 2 \end{bmatrix}$

Therefore

$3-x = 7 \Rightarrow x = -4$

$y+2 = 2 \Rightarrow y = 0$

2)    $\begin{bmatrix} -8 & x \end{bmatrix} + \begin{bmatrix} y & -2 \end{bmatrix} = \begin{bmatrix} -3 & 2 \end{bmatrix}$

Therefore

$-8+y=-3 \Rightarrow y = 5$

$x-2=2 \Rightarrow x = 4$

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Question 8: Given $M = \begin{bmatrix} 5 & -3 \\ -2 & 4 \end{bmatrix}$, find its transpose matrix $M^{t}$. If possible find:

1.  $M+M^{t}$
2.  $M^{t}-M$

$M = \begin{bmatrix} 5 & -3 \\ -2 & 4 \end{bmatrix}$

$M^{t} = \begin{bmatrix} 5 & -2 \\ -3 & 4 \end{bmatrix}$

1) $M+M^{t}$

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

2) $M^{t}-M$

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

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Question 9:  Write the additive  inverse of matrices A, B and C where $A = \begin{bmatrix} 6 & -5 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & 0 \\ 4 & -1 \end{bmatrix}$ and $C = \begin{bmatrix} -2 \\ 4 \end{bmatrix}$.

Additive Inverse of  $A = \begin{bmatrix} 6 & -5 \end{bmatrix}$ is $= \begin{bmatrix} -6 & 5 \end{bmatrix}$

Additive Inverse of $B = \begin{bmatrix} -2 & 0 \\ 4 & -1 \end{bmatrix}$ is $= \begin{bmatrix} 2 & 0 \\ -4 & 1 \end{bmatrix}$

Additive Inverse of $C = \begin{bmatrix} -2 \\ 4 \end{bmatrix}$  is $= \begin{bmatrix} 7 & -4 \end{bmatrix}$

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Question 10: Given $A = \begin{bmatrix} 2 & -3 \end{bmatrix}, \ B= \begin{bmatrix} 0 & 2 \end{bmatrix}, \ C= \begin{bmatrix} -1 & 4 \end{bmatrix}$. Find matrix $X$ in each of the following:

1.  $X+B=C-A$
2.  $A-X=B+C$

Let $X = \begin{bmatrix} a & b \end{bmatrix}$

1) $X+B=C-A$

$\begin{bmatrix} a & b \end{bmatrix} +\begin{bmatrix} 0 & 2 \end{bmatrix}=\begin{bmatrix} -1 & 4 \end{bmatrix}-\begin{bmatrix} 2 & -3 \end{bmatrix}$

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

Therefore

$a = -3 \ and \ b = 5$

Hence  $X = \begin{bmatrix} -3 & 5 \end{bmatrix}$

2) $A-X=B+C$

$\begin{bmatrix} 2 & -3 \end{bmatrix} - \begin{bmatrix} a & b \end{bmatrix} = \begin{bmatrix} 0 & 2 \end{bmatrix} + \begin{bmatrix} -1 & 4 \end{bmatrix}$

$\begin{bmatrix} 2-a & -3-b \end{bmatrix} = \begin{bmatrix} -1 & 6 \end{bmatrix}$

Therefore $a = 3 \ and \ b = -9$

Hence $X = \begin{bmatrix} 3 & -9 \end{bmatrix}$

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Question 11: Given $A = \begin{bmatrix} -1 & 0 \\ 2 & -4 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & -3 \\ -2 & 0 \end{bmatrix}$. FInd the matrix $X$  in each of the following:

1.  $A+X=B$
2.  $A-X=B$
3.  $X-B=A$

Let $X = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

1) $A+X=B$

$\begin{bmatrix} -1 & 0 \\ 2 & -4 \end{bmatrix}+\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} 3 & -3 \\ -2 & 0 \end{bmatrix}$

$\begin{bmatrix} -1+a & b \\ 2+c & -4+d \end{bmatrix} = \begin{bmatrix} 3 & -3 \\ -2 & 0 \end{bmatrix}$

Therefore $a = 4, \ b = -3, \ c = -4 \ and \ d = 4$

Hence $X = \begin{bmatrix} 4 & -3 \\ -4 & 4 \end{bmatrix}$

2) $A-X=B$

$\begin{bmatrix} -1 & 0 \\ 2 & -4 \end{bmatrix}-\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} 3 & -3 \\ -2 & 0 \end{bmatrix}$

$\begin{bmatrix} -1-a & -b \\ 2-c & -4-d \end{bmatrix} = \begin{bmatrix} 3 & -3 \\ -2 & 0 \end{bmatrix}$

Therefore $a = -4, \ b = 3, \ c = 4 \ and \ d = -4$

Hence $X = \begin{bmatrix} -4 & 3 \\ 4 & -4 \end{bmatrix}$

3) $X-B=A$

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} - \begin{bmatrix} 3 & -3 \\ -2 & 0 \end{bmatrix} =\begin{bmatrix} -1 & 0 \\ 2 & -4 \end{bmatrix}$

$\begin{bmatrix} a-3 & b+3 \\ c+2 & d \end{bmatrix} =\begin{bmatrix} -1 & 0 \\ 2 & -4 \end{bmatrix}$

Therefore $a = 2, \ b = -3, \ c = 0 \ and \ d = -4$

Hence $X = \begin{bmatrix} 2 & -3 \\ 0 & -4 \end{bmatrix}$

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