Determinants Class 12

Evaluate the determinants
$\left|\begin{array}{rr}2 & 4 \\ -5 & -1\end{array}\right|$

Evaluate the determinants
$\left|\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right|$ (ii) $\left|\begin{array}{cc}x^2-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$

Evaluate the determinants
$\left|\begin{array}{cc}x^2-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$

If $\mathrm{A}=\left[\begin{array}{ll}1 & 2 \\ 4 & 2\end{array}\right]$, then show that $|2 \mathrm{~A}|=4|\mathrm{~A}|$

If $\mathrm{A}=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{array}\right]$, then show that $|3 \mathrm{~A}|=27|\mathrm{~A}|$

Evaluate the determinants
(i) $\left|\begin{array}{rrr}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$

Evaluate the determinants
(ii) $\left|\begin{array}{rrr}3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1\end{array}\right|$

Evaluate the determinants
(iii) $\left|\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right|$

Evaluate the determinants
(iv) $\left|\begin{array}{rrr}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$

If $\mathrm{A}=\left[\begin{array}{rrr}1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9\end{array}\right]$, find $|\mathrm{A}|$

Find values of $x$, if
(i) $\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$

Find values of $x$, if
(ii) $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{cc}x & 3 \\ 2 x & 5\end{array}\right|$

If $\left|\begin{array}{cc}x & 2 \\ 18 & x\end{array}\right|=\left|\begin{array}{cc}6 & 2 \\ 18 & 6\end{array}\right|$, then $x$ is equal to
(A) 6
(B) $\pm 6$
(C) -6
(D) 0

Find area of the triangle with vertices at the point given in each of the following :
(i) (1, 0), (6, 0), $(4,3)$

Find area of the triangle with vertices at the point given in each of the following :
(ii) $(2,7),(1,1),(10,8)$

Find area of the triangle with vertices at the point given in each of the following :
(iii) (-2, -3), (3, 2), (-1, -8)

Show that points
$\mathrm{A}(a, b+c), \mathrm{B}(b, c+a), \mathrm{C}(c, a+b)$ are collinear.

Find values of $k$ if area of triangle is 4 sq. units and vertices are
(i) $(k, 0),(4,0),(0,2)$

Find values of $k$ if area of triangle is 4 sq. units and vertices are
(ii) $(-2,0),(0,4),(0, k)$

Find equation of line joining $(1,2)$ and $(3,6)$ using determinants.

Find equation of line joining $(3,1)$ and $(9,3)$ using determinants.

If area of triangle is 35 sq units with vertices (2, -6), $(5,4)$ and $(k, 4)$. Then $k$ is
(A) 12
(B) -2
(C) -12, -2
(D) $12,-2$

Write Minors and Cofactors of the elements of following determinants:
$\left|\begin{array}{rr}2 & -4 \\ 0 & 3\end{array}\right|$

Write Minors and Cofactors of the elements of following determinants:
$\left|\begin{array}{cc}a & c \\ b & d\end{array}\right|$

$\left|\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|$

$\left|\begin{array}{rrr}1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2\end{array}\right|$

Using Cofactors of elements of second row, evaluate $\Delta=\left|\begin{array}{lll}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{array}\right|$

Using Cofactors of elements of third column, evaluate $\Delta=\left|\begin{array}{lll}1 & x & y z \\ 1 & y & z x \\ 1 & z & x y\end{array}\right|$.

If $\Delta=\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$ and $\mathrm{A}_{i j}$ is Cofactors of $a_{i j}$, then value of $\Delta$ is given by
(A) $a_{11} \mathrm{~A}_{31}+a_{12} \mathrm{~A}_{32}+a_{13} \mathrm{~A}_{33}$
(B) $a_{11} \mathrm{~A}_{11}+a_{12} \mathrm{~A}_{21}+a_{13} \mathrm{~A}_{31}$
(C) $a_{21} \mathrm{~A}_{11}+a_{22} \mathrm{~A}_{12}+a_{23} \mathrm{~A}_{13}$
(D) $a_{11} \mathrm{~A}_{11}+a_{21} \mathrm{~A}_{21}+a_{31} \mathrm{~A}_{31}$

Find adjoint of each of the matrices
$\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$

$\left[\begin{array}{ccc}1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1\end{array}\right]$

Verify $\mathrm{A}(\operatorname{adj} \mathrm{A})=(\operatorname{adj} \mathrm{A}) \mathrm{A}=|\mathrm{A}| \mathrm{I}$
$\left[\begin{array}{cc}2 & 3 \\ -4 & -6\end{array}\right]$

Verify $\mathrm{A}(\operatorname{adj} \mathrm{A})=(\operatorname{adj} \mathrm{A}) \mathrm{A}=|\mathrm{A}| \mathrm{I}$
$\left[\begin{array}{ccc}1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3\end{array}\right]$

Find the inverse of each of the matrices (if it exists) given
$\left[\begin{array}{cc}2 & -2 \\ 4 & 3\end{array}\right]$

Find the inverse of each of the matrices (if it exists) given
$\left[\begin{array}{ll}-1 & 5 \\ -3 & 2\end{array}\right]$

Find the inverse of each of the matrices (if it exists) given
$\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5\end{array}\right]$

Find the inverse of each of the matrices (if it exists) given
$\left[\begin{array}{ccc}1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1\end{array}\right]$

Find the inverse of each of the matrices (if it exists) given
$\left[\begin{array}{ccc}2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1\end{array}\right]$

$\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]$

$\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \alpha & \sin \alpha \\ 0 & \sin \alpha & -\cos \alpha\end{array}\right]$

Let $\mathrm{A}=\left[\begin{array}{ll}3 & 7 \\ 2 & 5\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}6 & 8 \\ 7 & 9\end{array}\right]$. Verify that $(\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1}$.

If $\mathrm{A}=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]$, show that $\mathrm{A}^2-5 \mathrm{~A}+7 \mathrm{I}=\mathrm{O}$. Hence find $\mathrm{A}^{-1}$.

For the matrix $\mathrm{A}=\left[\begin{array}{ll}3 & 2 \\ 1 & 1\end{array}\right]$, find the numbers $a$ and $b$ such that $\mathrm{A}^2+a \mathrm{~A}+b \mathrm{I}=\mathrm{O}$.

For the matrix $\mathrm{A}=\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3\end{array}\right]$
Show that $\mathrm{A}^3-6 \mathrm{~A}^2+5 \mathrm{~A}+11 \mathrm{I}=\mathrm{O}$. Hence, find $\mathrm{A}^{-1}$.

If $\mathrm{A}=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right]$
Verify that $\mathrm{A}^3-6 \mathrm{~A}^2+9 \mathrm{~A}-4 \mathrm{I}=\mathrm{O}$ and hence find $\mathrm{A}^{-1}$

Let A be a nonsingular square matrix of order 3 × 3. Then $|\operatorname{adj} \mathrm{A}|$ is equal to
(A) $|\mathrm{A}|$
(B) $|\mathrm{A}|^2$
(C) $|\mathrm{A}|^3$
(D) $3|\mathrm{~A}|$

If A is an invertible matrix of order 2, then $\operatorname{det}\left(\mathrm{A}^{-1}\right)$ is equal to
(A) $\operatorname{det}(\mathrm{A})$
(B) $\frac{1}{\operatorname{det}(\mathrm{~A})}$
(C) 1
(D) 0

Examine the consistency of the system of equations.
$$
\begin{aligned}
& x+2 y=2 \\
& 2 x+3 y=3
\end{aligned}
$$

Examine the consistency of the system of equations.
$$
\begin{aligned}
& 2 x-y=5 \\
& x+y=4
\end{aligned}
$$

Examine the consistency of the system of equations.
$$
\begin{aligned}
& x+3 y=5 \\
& 2 x+6 y=8
\end{aligned}
$$

Examine the consistency of the system of equations.
$$
\begin{aligned}
& x+y+z=1 \\
& 2 x+3 y+2 z=2 \\
& a x+a y+2 a z=4
\end{aligned}
$$

Examine the consistency of the system of equations.
$$
\begin{aligned}
& 3 x-y-2 z=2 \\
& 2 y-z=-1 \\
& 3 x-5 y=3
\end{aligned}
$$

Examine the consistency of the system of equations.
$$
\begin{aligned}
& 5 x-y+4 z=5 \\
& 2 x+3 y+5 z=2 \\
& 5 x-2 y+6 z=-1
\end{aligned}
$$

Solve system of linear equations, using matrix method.
$$
\begin{aligned}
& 5 x+2 y=4 \\
& 7 x+3 y=5
\end{aligned}
$$

Solve system of linear equations, using matrix method.
$$
\begin{aligned}
& 2 x-y=-2 \\
& 3 x+4 y=3
\end{aligned}
$$

Solve system of linear equations, using matrix method.
$$
\begin{aligned}
& 4 x-3 y=3 \\
& 3 x-5 y=7
\end{aligned}
$$

Solve system of linear equations, using matrix method.
$$
\begin{aligned}
& 5 x+2 y=3 \\
& 3 x+2 y=5
\end{aligned}
$$

Solve system of linear equations, using matrix method.
$$
\begin{gathered}
2 x+y+z=1 \\
x-2 y-z=\frac{3}{2} \\
3 y-5 z=9
\end{gathered}
$$

12.

$$
\begin{gathered}
x-y+z=4 \\
2 x+y-3 z=0 \\
x+y+z=2
\end{gathered}
$$

13.

$$
\begin{aligned}
& 2 x+3 y+3 z=5 \\
& x-2 y+z=-4 \\
& 3 x-y-2 z=3
\end{aligned}
$$

14.

$$
\begin{aligned}
& x-y+2 z=7 \\
& 3 x+4 y-5 z=-5 \\
& 2 x-y+3 z=12
\end{aligned}
$$

Solve system of linear equations, using matrix method.
$$
\begin{gathered}
x-y+z=4 \\
2 x+y-3 z=0 \\
x+y+z=2
\end{gathered}
$$

Solve system of linear equations, using matrix method.
$$
\begin{aligned}
& 2 x+3 y+3 z=5 \\
& x-2 y+z=-4 \\
& 3 x-y-2 z=3
\end{aligned}
$$

Solve system of linear equations, using matrix method.
$$
\begin{aligned}
& x-y+2 z=7 \\
& 3 x+4 y-5 z=-5 \\
& 2 x-y+3 z=12
\end{aligned}
$$

If $\mathrm{A}=\left[\begin{array}{rrr}2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2\end{array}\right]$, find $\mathrm{A}^{-1}$. Using $\mathrm{A}^{-1}$ solve the system of equations

$$
\begin{aligned}
2 x-3 y+5 z & =11 \\
3 x+2 y-4 z & =-5 \\
x+y-2 z & =-3
\end{aligned}
$$

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by
matrix method.

Evaluate $\left|\begin{array}{rr}2 & 4 \\ -1 & 2\end{array}\right|$.

Evaluate $\left|\begin{array}{cc}x & x+1 \\ x-1 & x\end{array}\right|$

Evaluate the determinant $\Delta=\left|\begin{array}{rrr}1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0\end{array}\right|$.

Evaluate $\Delta=\left|\begin{array}{ccc}0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$.

Find values of $x$ for which $\left|\begin{array}{ll}3 & x \\ x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|$.

Find the area of the triangle whose vertices are $(3,8),(-4,2)$ and $(5,1)$.

Find the equation of the line joining $\mathrm{A}(1,3)$ and $\mathrm{B}(0,0)$ using determinants and find $k$ if $\mathrm{D}(k, 0)$ is a point such that area of triangle ABD is 3 sq units.

Find the minor of element 6 in the determinant $\Delta=\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right|$

Find minors and cofactors of all the elements of the determinant $\left|\begin{array}{rr}1 & -2 \\ 4 & 3\end{array}\right|$

Find minors and cofactors of the elements $a_{11}, a_{21}$ in the determinant

$$
\Delta=\left|\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|
$$

Find minors and cofactors of the elements of the determinant $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ and verify that $a_{11} \mathrm{~A}_{31}+a_{12} \mathrm{~A}_{32}+a_{13} \mathrm{~A}_{33}=0$

Find $\operatorname{adj} \mathrm{A}$ for $\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right]$

If $\mathrm{A}=\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right]$, then verify that $\mathrm{A} \operatorname{adj} \mathrm{A}=|\mathrm{A}| \mathrm{I}$. Also find $\mathrm{A}^{-1}$.

If $\mathrm{A}=\left[\begin{array}{cc}2 & 3 \\ 1 & -4\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}1 & -2 \\ -1 & 3\end{array}\right]$, then verify that $(\mathrm{AB})^{-1}=\mathrm{B}^{-1} \mathrm{~A}^{-1}$.

Show that the matrix $\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$ satisfies the equation $\mathrm{A}^2-4 \mathrm{~A}+\mathrm{I}=\mathrm{O}$, where I is $2 \times 2$ identity matrix and O is $2 \times 2$ zero matrix. Using this equation, find $\mathrm{A}^{-1}$.

Solve the system of equations

$$
\begin{aligned}
& 2 x+5 y=1 \\
& 3 x+2 y=7
\end{aligned}
$$

Solve the following system of equations by matrix method.

$$
\begin{array}{r}
3 x-2 y+3 z=8 \\
2 x+y-z=1 \\
4 x-3 y+2 z=4
\end{array}
$$

The sum of three numbers is 6. If we multiply third number by 3 and add second number to it, we get 11. By adding first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method.

Use product $\left[\begin{array}{ccc}1 & 1 & 2 \\ 0 & 2 & 3 \\ 3 & 2 & 4\end{array}\right]\left[\begin{array}{ccc}2 & 0 & 1 \\ 9 & 2 & 3 \\ 6 & 1 & 2\end{array}\right]$ to solve the system of equations

$$
\begin{array}{r}
x-y+2 z=1 \\
2 y-3 z=1 \\
3 x-2 y+4 z=2
\end{array}
$$

Prove that the determinant $\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$ is independent of $\theta$.

Evaluate $\left|\begin{array}{ccc}\cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin \alpha \\ -\sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{array}\right|$

If $A^{-1}=\left[\begin{array}{ccc}3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1\end{array}\right]$, find $(A B)^{-1}$

Let $\mathrm{A}=\left[\begin{array}{ccc}1 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 5\end{array}\right]$. Verify that
(i) $[\operatorname{adj} \mathrm{A}]^{-1}=\operatorname{adj}\left(\mathrm{A}^{-1}\right)$
(ii) $\left(\mathrm{A}^{-1}\right)^{-1}=\mathrm{A}$

Evaluate $\left|\begin{array}{ccc}x & y & x+y \\ y & x+y & x \\ x+y & x & y\end{array}\right|$

Evaluate $\left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$

Solve the system of equations

$$
\begin{aligned}
& \frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4 \\
& \frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1 \\
& \frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2
\end{aligned}
$$

Choose the correct answer.
If $x, y, z$ are nonzero real numbers, then the inverse of matrix $\mathrm{A}=\left[\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$ is
(A) $\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$
(B) $x y z\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$
(C) $\frac{1}{x y z}\left[\begin{array}{ccc}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$
(D) $\frac{1}{x y z}\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$

Choose the correct answer.
Let $\mathrm{A}=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]$, where $0 \leq \theta \leq 2 \pi$. Then
(A) $\operatorname{Det}(\mathrm{A})=0$
(B) $\operatorname{Det}(\mathrm{A}) \in(2, \infty)$
(C) $\operatorname{Det}(\mathrm{A}) \in(2,4)$
(D) $\operatorname{Det}(\mathrm{A}) \in[2,4]$