1 Marks Question

Question 11 Mark

For which value of $x,\left|\begin{array}{ll}3 & 2 \\ 5 & x\end{array}\right|$ will be zero?

Answer

$
\left|\begin{array}{ll}
3 & 2 \\
5 & x
\end{array}\right|=0
$
$\Rightarrow 3 x-10=0 \Rightarrow x=\frac{10}{3} \quad$

Question 21 Mark

In determinant $\left|\begin{array}{lll}1 & 3 & 2 \\ 8 & 6 & 3 \\ 9 & 5 & 4\end{array}\right|$ find the minor of element 6.

Answer

$
M_{22}=\left|\begin{array}{ll}
1 & 2 \\
9 & 4
\end{array}\right|
$

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Question 31 Mark

If $\left|\begin{array}{ccc}\lambda+1 & 1 & 1 \\ 1 & 1 & -1 \\ -1 & 1 & 1\end{array}\right|=4$ then find the value of $\lambda$.

Answer

$(\lambda+1)(1+1)-1(1-1)+1(1+1)=4$
$\Rightarrow 2 \lambda+2-0+2=4$
$\Rightarrow \lambda=0$

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Question 41 Mark

Find value of determinant $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+a & 1 \\ 1 & 1 & 1+b\end{array}\right|$.

Answer

$=1\{(1+a)(1+b)-1\}-1(1+b-1)+1(1-1-a)$
$=1+a+b+a b-1-b-a$
$=a b$

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Question 51 Mark

If $\left|\begin{array}{cc}3 x & 7 \\ -2 & 4\end{array}\right|=\left|\begin{array}{ll}8 & 7 \\ 6 & 4\end{array}\right|$ then find the value of $x$.

Answer

$
\begin{aligned}
& & \left|\begin{array}{ll}
3 x & 7 \\
-2 & 4
\end{array}\right| & =\left|\begin{array}{ll}
8 & 7 \\
6 & 4
\end{array}\right| \\
\Rightarrow & & 12 x+14 & =32-42 \\
\Rightarrow & & 12 x & =-24 \\
\Rightarrow & & x & =-2
\end{aligned}
$

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Question 61 Mark

If $A$ and $B$ has two $n$ order invertible matrix. Then find value of $( AB )^{-1}$.

Answer

$( AB )^{-1}= B ^{-1} A^{-1}$.

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Question 71 Mark

Write the matrix form $\left[\begin{array}{lll}5 & 3 & 1 \\ 2 & 1 & 3 \\ 1 & 2 & 4\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}16 \\ 19 \\ 25\end{array}\right]$ is system of equations.

Answer

$5 x+3 y+z=16 , 2 x+y+3 z=19 , x+2 y+4 z=25$

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Question 81 Mark

If $A =\left[\begin{array}{cc}3 & -3 \\ 3 & 3\end{array}\right], B =\left[\begin{array}{ll}6 & 3 \\ 4 & 2\end{array}\right]$ and $C =\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ then which of the matrix will be invertible matrix?

Answer

Here $|A| \neq 0,|B|=0,|C| \neq 0$
Hence invertible matrix will be$
A=\left[\begin{array}{cc}
3 & -3 \\
3 & 3
\end{array}\right] \text { and } C=\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]
$

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Question 91 Mark

If $A=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]$, find the value of $A+A^{\prime}$.

Answer

$
\begin{aligned}
A+A^{\prime} & =\left[\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right]+\left[\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right]^{\prime} \\
& =\left[\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right]+\left[\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right]=\left[\begin{array}{ll}
2 & 4 \\
4 & 2
\end{array}\right]
\end{aligned}
$

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Question 101 Mark

If in any determinant, any two rows or two columns has all same elements, then find the value of determinant.

Answer

The value of determinant will be zero.

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Question 111 Mark

Prove that $\left|\begin{array}{cc}x^2+x+1 & x+1 \\ x & x-1\end{array}\right| =x^3-x^2-x-1$

Answer

Suppose that $\Delta=\left|\begin{array}{cc}x^2+x+1 & x+1 \\ x & x-1\end{array}\right|$
expanding the determinant
$\Delta=\left(x^2+x+1\right)(x-1)-(x+1) x$
$=x^3-1-x^2-x$
$=x^3-x^2-x-1 \text { Hence proved. }$

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Question 121 Mark

Represent the following system of equations in the form of matrix equation $:x+y+z=6 , x-y+z=2 , 2 x+y-z=1$

Answer

$\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -1 & 1 \\ 2 & 1 & -1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}6 \\ 2 \\ 1\end{array}\right]$

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