Evaluate the determinant:
\[ \left| \begin{matrix} 2 & 4 \\ -5 & -1 \end{matrix} \right| \]
18
Evaluate the determinants:
(i) \[ \left| \begin{matrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{matrix} \right| \]
(i) 1
(ii) \(x^3 - x^2 + 2\)
If \(A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}\), then show that \(|2A| = 4|A|\).
If \(A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}\), then show that \(|3A| = 27|A|\).
Evaluate the determinants:
(i) \[ \left| \begin{matrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{matrix} \right| \]
(iii) \[ \left| \begin{matrix} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{matrix} \right| \]
(iv) \[ \left| \begin{matrix} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{matrix} \right| \]
(i) -12
(ii) 46
(iii) 0
(iv) 5
If \(A = \begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}\), find \(|A|\).
0
Find values of \(x\), if
(i) \[ \left| \begin{matrix} 2 & 4 \\ 5 & 1 \end{matrix} \right| = \left| \begin{matrix} 2x & 4 \\ 6 & x \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix} \right| = \left| \begin{matrix} x & 3 \\ 2x & 5 \end{matrix} \right| \]
(i) \(x = \pm \sqrt{3}\)
(ii) \(x = 2\)
If
\[ \left| \begin{matrix} x & 2 \\ 18 & x \end{matrix} \right| = \left| \begin{matrix} 6 & 2 \\ 18 & 6 \end{matrix} \right| \]
then \(x\) is equal to
(A) 6
(B) \(\pm 6\)
(C) -6
(D) 0
(B)
Find area of the triangle with vertices at the point given in each of the following:
(i) \((1,0), (6,0), (4,3)\)
(ii) \((2,7), (1,1), (10,8)\)
(iii) \((-2,-3), (3,2), (-1,-8)\)
(i) \(\frac{15}{2}\)
(ii) \(\frac{47}{2}\)
(iii) 15
Show that points \(A(a, b+c),\; B(b, c+a),\; C(c, a+b)\) are collinear.
Find values of \(k\) if area of triangle is 4 sq. units and vertices are
(i) \((k,0), (4,0), (0,2)\)
(ii) \((-2,0), (0,4), (0,k)\)
(i) \(k = 0,\; 8\)
(ii) \(k = 0,\; 8\)
(i) Find equation of line joining \((1,2)\) and \((3,6)\) using determinants.
(ii) Find equation of line joining \((3,1)\) and \((9,3)\) using determinants.
(i) \(y = 2x\)
(ii) \(x - 3y = 0\)
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
(D)
Write minors and cofactors of the elements of following determinants:
(i) \[ \left| \begin{matrix} 2 & -4 \\ 0 & 3 \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} a & c \\ b & d \end{matrix} \right| \]
(i) \(M_{11}=3,\; M_{12}=0,\; M_{21}=-4,\; M_{22}=2,\; A_{11}=3,\; A_{12}=0,\; A_{21}=4,\; A_{22}=2\)
(ii) \(M_{11}=d,\; M_{12}=b,\; M_{21}=c,\; M_{22}=a\)
\(A_{11}=d,\; A_{12}=-b,\; A_{21}=-c,\; A_{22}=a\)
Write minors and cofactors of the elements of following determinants:
(i) \[ \left| \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right| \]
(ii) \[ \left| \begin{matrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{matrix} \right| \]
(i) \(M_{11}=1,\; M_{12}=0,\; M_{13}=0,\; M_{21}=0,\; M_{22}=1,\; M_{23}=0,\; M_{31}=0,\; M_{32}=0,\; M_{33}=1\)
\(A_{11}=1,\; A_{12}=0,\; A_{13}=0,\; A_{21}=0,\; A_{22}=1,\; A_{23}=0,\; A_{31}=0,\; A_{32}=0,\; A_{33}=1\)
(ii) \(M_{11}=11,\; M_{12}=6,\; M_{13}=3,\; M_{21}=-4,\; M_{22}=2,\; M_{23}=1,\; M_{31}=-20,\; M_{32}=-13,\; M_{33}=5\)
\(A_{11}=11,\; A_{12}=-6,\; A_{13}=3,\; A_{21}=4,\; A_{22}=2,\; A_{23}=-1,\; A_{31}=-20,\; A_{32}=13,\; A_{33}=5\)
Using cofactors of elements of second row, evaluate
\[ \Delta = \left| \begin{matrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{matrix} \right| \]
7
Using cofactors of elements of third column, evaluate
\[ \Delta = \left| \begin{matrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{matrix} \right| \]
\((x-y)(y-z)(z-x)\)
If
\[ \Delta = \left| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right| \]
and \(A_{ij}\) is cofactor of \(a_{ij}\), then value of \(\Delta\) is given by
Find adjoint of the matrix
\[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \]
\(\operatorname{adj}A = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}\)
Find adjoint of the matrix
\[ \begin{bmatrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{bmatrix} \]
\(\operatorname{adj}A = \begin{bmatrix} 3 & 1 & -11 \\ -12 & 5 & -1 \\ 6 & 2 & 5 \end{bmatrix}\)
Verify that \(A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|I\) for
\[ A = \begin{bmatrix} 2 & 3 \\ -4 & -6 \end{bmatrix} \]
Verify that \(A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|I\) for
\[ A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix} \]
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix} \]
\(A^{-1} = \frac{1}{14}\begin{bmatrix} 3 & 2 \\ -4 & 2 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix} \]
\(A^{-1} = \frac{1}{13}\begin{bmatrix} 2 & -5 \\ 3 & -1 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix} \]
\(A^{-1} = \frac{1}{10}\begin{bmatrix} 10 & -10 & 2 \\ 0 & 5 & -4 \\ 0 & 0 & 2 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix} \]
\(A^{-1} = -\frac{1}{3}\begin{bmatrix} -3 & 0 & 0 \\ 3 & -1 & 0 \\ -9 & -2 & 3 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1 \end{bmatrix} \]
\(A^{-1} = -\frac{1}{3}\begin{bmatrix} -1 & 5 & 3 \\ -4 & 23 & 12 \\ 1 & -11 & -6 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix} \]
\(A^{-1} = \begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix}\)
Find the inverse of the matrix (if it exists)
\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha \\ 0 & \sin\alpha & -\cos\alpha \end{bmatrix} \]
\(A^{-1} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha \\ 0 & \sin\alpha & -\cos\alpha \end{bmatrix}\)
Let \(A = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix}\) and \(B = \begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}\). Verify that \((AB)^{-1} = B^{-1}A^{-1}\).
If \(A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}\), show that \(A^2 - 5A + 7I = O\). Hence, find \(A^{-1}\).
\(A^{-1} = \frac{1}{7}\begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix}\)
For the matrix \(A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}\), find the numbers \(a\) and \(b\) such that \(A^2 + aA + bI = O\).
\(a = -4,\; b = 1\)
For the matrix
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 2 & -1 & 3 \end{bmatrix} \]
show that \(A^3 - 6A^2 + 5A + 11I = O\). Hence, find \(A^{-1}\).
\(A^{-1} = \frac{1}{11}\begin{bmatrix} -3 & 4 & 5 \\ 9 & -1 & -4 \\ 5 & -3 & -1 \end{bmatrix}\)
If
\[ A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix} \]
verify that \(A^3 - 6A^2 + 9A - 4I = O\) and hence find \(A^{-1}\).
\(A^{-1} = \frac{1}{4}\begin{bmatrix} 3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix}\)
Let \(A\) be a nonsingular square matrix of order \(3 \times 3\). Then \(|\operatorname{adj}A|\) is equal to
If \(A\) is an invertible matrix of order 2, then \(\det(A^{-1})\) is equal to
Examine the consistency of the system of equations:
\(x + 2y = 2\)
\(2x + 3y = 3\)
Consistent
Examine the consistency of the system of equations:
\(2x - y = 5\)
\(x + y = 4\)
Consistent
Examine the consistency of the system of equations:
\(x + 3y = 5\)
\(2x + 6y = 8\)
Inconsistent
Examine the consistency of the system of equations:
\(x + y + z = 1\)
\(2x + 3y + 2z = 2\)
\(ax + ay + 2az = 4\)
Consistent
Examine the consistency of the system of equations:
\(3x - y - 2z = 2\)
\(2y - z = -1\)
\(3x - 5y = 3\)
Inconsistent
Examine the consistency of the system of equations:
\(5x - y + 4z = 5\)
\(2x + 3y + 5z = 2\)
\(5x - 2y + 6z = -1\)
Consistent
Solve the system of linear equations, using matrix method:
\(5x + 2y = 4\)
\(7x + 3y = 5\)
\(x = 2,\; y = -3\)
Solve the system of linear equations, using matrix method:
\(2x - y = -2\)
\(3x + 4y = 3\)
\(x = -\frac{5}{11},\; y = \frac{12}{11}\)
Solve the system of linear equations, using matrix method:
\(4x - 3y = 3\)
\(3x - 5y = 7\)
\(x = -\frac{6}{11},\; y = -\frac{19}{11}\)
Solve the system of linear equations, using matrix method:
\(5x + 2y = 3\)
\(3x + 2y = 5\)
\(x = -1,\; y = 4\)
Solve the system of linear equations, using matrix method:
\(2x + y + z = 1\)
\(x - 2y - z = \frac{3}{2}\)
\(3y - 5z = 9\)
\(x = 1,\; y = \frac{1}{2},\; z = -\frac{3}{2}\)
Solve the system of linear equations, using matrix method:
\(x - y + z = 4\)
\(2x + y - 3z = 0\)
\(x + y + z = 2\)
\(x = 2,\; y = -1,\; z = 1\)
Solve the system of linear equations, using matrix method:
\(2x + 3y + 3z = 5\)
\(x - 2y + z = -4\)
\(3x - y - 2z = 3\)
\(x = 1,\; y = 2,\; z = -1\)
Solve the system of linear equations, using matrix method:
\(x - y + 2z = 7\)
\(3x + 4y - 5z = -5\)
\(2x - y + 3z = 12\)
\(x = 2,\; y = 1,\; z = 3\)
If
\[ A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix} \]
find \(A^{-1}\). Using \(A^{-1}\) solve the system of equations
\(2x - 3y + 5z = 11\)
\(3x + 2y - 4z = -5\)
\(x + y - 2z = -3\)
\(A^{-1} = \begin{bmatrix} 0 & 1 & -2 \\ -2 & 9 & -23 \\ -1 & 5 & -13 \end{bmatrix}\), \(x = 1,\; y = 2,\; z = 3\)
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is \(\text{₹}60\). The cost of 2 kg onion, 4 kg wheat and 6 kg rice is \(\text{₹}90\). The cost of 6 kg onion 2 kg wheat and 3 kg rice is \(\text{₹}70\). Find cost of each item per kg by matrix method.
Cost of onions per kg = \(\text{₹}5\)
Cost of wheat per kg = \(\text{₹}8\)
Cost of rice per kg = \(\text{₹}8\)
Prove that the determinant
\[ \left| \begin{matrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{matrix} \right| \]
is independent of \(\theta\).
Evaluate
\[ \left| \begin{matrix} \cos\alpha\cos\beta & \cos\alpha\sin\beta & -\sin\alpha \\ -\sin\beta & \cos\beta & 0 \\ \sin\alpha\cos\beta & \sin\alpha\sin\beta & \cos\alpha \end{matrix} \right| \]
1
If
\[ A^{-1} = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}, \]
find \((AB)^{-1}\).
\((AB)^{-1} = \begin{bmatrix} 9 & -3 & 5 \\ -2 & 1 & 0 \\ 1 & 0 & 2 \end{bmatrix}\)
Let
\[ A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix}. \]
Verify that
(i) \([\operatorname{adj}A]^{-1} = \operatorname{adj}(A^{-1})\)
(ii) \((A^{-1})^{-1} = A\)
Evaluate
\[ \left| \begin{matrix} x & y & x+y \\ y & x+y & x \\ x+y & x & y \end{matrix} \right| \]
\(-2(x^3 + y^3)\)
Evaluate
\[ \left| \begin{matrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{matrix} \right| \]
\(xy\)
Solve the system of equations:
\(\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4\)
\(\frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1\)
\(\frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2\)
\(x = 2,\; y = 3,\; z = 5\)
Choose the correct answer.
If \(x, y, z\) are nonzero real numbers, then the inverse of matrix
\[ A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} \]
is
Let
\[ A = \begin{bmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{bmatrix}, \quad 0 \le \theta \le 2\pi. \]
Then