2 Marks

Question 12 Marks

If $\text{A}=\begin{bmatrix}2&4\\4&3\end{bmatrix},\text{X}=\begin{bmatrix}\text{n}\\1\end{bmatrix},\text{B}=\begin{bmatrix}8\\11\end{bmatrix}$and AX = B, then find n.

Answer

Here,
$\begin{bmatrix}2&4\\4&3\end{bmatrix}\begin{bmatrix}\text{n}\\1\end{bmatrix}=\begin{bmatrix}8\\11\end{bmatrix}$
$\Rightarrow\begin{bmatrix}2\text{n}+4\\4\text{n}+3\end{bmatrix}=\begin{bmatrix}8\\11\end{bmatrix}$
$\Rightarrow2\text{n}+4=8$
$\Rightarrow2\text{n}=4$
$\Rightarrow\text{n}=2$

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Question 22 Marks

If $\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}2\\-1\\3\end{bmatrix}$, find x, y, z.

Answer

$\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}2\\-1\\3\end{bmatrix}$
$\begin{bmatrix}\text{x}+0+0\\0+0+\text{z}\\0+\text{y}+0\end{bmatrix}=\begin{bmatrix}2\\-1\\3\end{bmatrix}$
$\begin{bmatrix}\text{x}\\\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}2\\-1\\3\end{bmatrix}$
$\therefore\ \text{x}=2,\ \text{y}=3\text{ and }\text{z}=-1$

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Question 32 Marks

If $\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}1\\0\\1\end{bmatrix}$, find x, y and z.

Answer

Here,
$\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}1\\0\\1\end{bmatrix}$
$\begin{bmatrix}\text{x}+0+0\\0-\text{y}+0\\0+0-\text{z}\end{bmatrix}=\begin{bmatrix}1\\0\\1\end{bmatrix}$
$\begin{bmatrix}\text{x}\\-\text{y}\\-\text{z}\end{bmatrix}=\begin{bmatrix}1\\0\\1\end{bmatrix}$
Hence, $\text{x}=1,\text{y}=0\text{ and }\text{z}=-1$

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Question 42 Marks

If $\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}1\\-1\\0\end{bmatrix}$, find x, y and z.

Answer

Here,
$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}1\\-1\\0\end{bmatrix}$
$\Rightarrow\begin{bmatrix}\text{x}\\\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}1\\-1\\0\end{bmatrix}$
$\therefore\ \text{x}=1,\ \text{y}=-1\text{ and }\text{z}=0$

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Question 52 Marks

If $\begin{bmatrix}1&0&0\\0&\text{y}&0\\0&0&1\end{bmatrix}\begin{bmatrix}\text{x}\\-1\\\text{z}\end{bmatrix}=\begin{bmatrix}1\\0\\1\end{bmatrix}$, find x, y and z.

Answer

Here,
$\begin{bmatrix}1&0&0\\0&\text{y}&0\\0&0&1\end{bmatrix}\begin{bmatrix}\text{x}\\-1\\\text{z}\end{bmatrix}=\begin{bmatrix}1\\0\\1\end{bmatrix}$
$\Rightarrow\begin{bmatrix}\text{x}\\-\text{y}\\\text{z}\end{bmatrix}=\begin{bmatrix}1\\0\\1\end{bmatrix}$
$\therefore\ \text{x}=1,\text{y}=0\text{ and }\text{z}=1$

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Maths 2 Marks

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