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If [latex]\begin{bmatrix} x-y-z\\ -y+z\\ z \end{bmatrix}=\begin{bmatrix} 0\\ 5\\ 3 \end{bmatrix}\\[/latex] then x, y and z are

1) 5 , 2 , 2 2) 1 , – 2 , 3 3) 0 , – 3 , 3... View Article

If the system of equations x – ky – z = 0, kx – y – z = 0, x + y – z = 0 has a non-zero solution, then the possible values of k are

1) –1, 2 2) 1, 2 3) 0, 1 4) –1, 1 Solution: (4) –1, 1 \(\begin{array}{l}\begin{array}{l} \therefore\left|\begin{array}{ccc} 1 &... View Article

If [latex]A=\begin{bmatrix} 3 &x-1 \\ 2x+3&x+2 \end{bmatrix}[/latex] is a symmetric matrix, then the value of x is

1) 4 2) 3 3) – 4 4) – 3 Solution: (3) – 4 \(\begin{array}{l}\begin{array}{l} A=\left[\begin{array}{cc} 3 & x-1 \\... View Article

If A = [latex]\begin{bmatrix} 1 &2 &2 \\ 2&1 &-2 \\ a&2 &b \end{bmatrix}[/latex] is a matrix satisfying AA^T = 9I₃ then the values of a and b are

1) 1 , 2 2) – 1 , 2 3) – 1 , – 2 4) 2 , 1 5)... View Article

[latex]A=\begin{bmatrix} 1 &0 &0 \\ 0&1 &1 \\ 0&-2 &4 \end{bmatrix},I=\begin{bmatrix} 1 &0 &0 \\ 0&1 &0 \\ 0&0 &1 \end{bmatrix},A^{-1}=[(1/6)(A^{2}+cA+dI)][/latex], then the value of c and d are

1) (–6, –11) 2) (6, 11) 3) (–6, 11) 4) (6, –11) Solution: (3) (–6, 11) \(\begin{array}{l}6A^{-1} =A^2 + cA... View Article

If A and B are two symmetric matrices of order 3. Statement I A(BA) and (AB) A are symmetric matrices. Statement II AB is a symmetric matrix If matrix multiplication of A and B is commutative.

1) Statement I is correct, Statement II is correct; Statement II is not a correct explanation for the statement I... View Article

If M is a 3 × 3 matrix satisfying [latex]M\begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix}=\begin{bmatrix} -1\\ 2\\ 3 \end{bmatrix},M\begin{bmatrix} 1\\ -1\\ 0 \end{bmatrix}=\begin{bmatrix} 1\\ 1\\ -1 \end{bmatrix},M\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}=\begin{bmatrix} 0\\ 0\\ 12 \end{bmatrix}[/latex], then the sum of the diagonal entries of M is

1) 9 2) 8 3) 10 4) 11 Solution: (1) 9 \(\begin{array}{l}\begin{aligned} &\text { Let } M=\left[\begin{array}{lll} a_{1} & a_{2}... View Article

The parameter, on which the value of the determinant [latex]\begin{vmatrix} 1 &a &a^{2} \\ cos (p-d)x&cospx &cos(p+d)x \\ sin (p-d)x&sinpx &sin(p+d)x \end{vmatrix}[/latex] does not depend upon is

1) a 2) p 3) d 4) x Solution: 2) p \(\begin{array}{l}\text { Let } \Delta=\left|\begin{array}{ccc} 1 & a &... View Article

If ω is a complex cube root of unity with ω ≠ 1 and p = [pij] is a n × n matrix with P_ij = ω^{i + j}. Then P² ≠ 0 when n is equal to

1) 55 2) 58 3) 56 4) All of these Solution: (4) All of these P = [Pij]nxm; Pij =... View Article

If P is a 3 × 3 matrix such that P^T = 2P + I, where P^T is the transpose of P and I is the 3 × 3 identity matrix, then there exists a column matrix, [latex]X=\begin{bmatrix} x\\ y\\ z \end{bmatrix}\neq\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}[/latex] such that

1) \(\begin{array}{l}PX=\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}\end{array} \) 2) PX 3) PX = 2X 4) PX = – X Solution: (4)... View Article

Compute [latex]\begin{bmatrix} 1 &3 &0 \\ 2&1 &4 \end{bmatrix} \times \begin{bmatrix} 1 &3 \\ 0&2 \\ -2&-1 \end{bmatrix}[/latex]

1) \(\begin{array}{l}\begin{bmatrix} 1 &3 \\ 6&-4 \end{bmatrix}\end{array} \) 2) \(\begin{array}{l}\begin{bmatrix} -1 &-3 \\ 6&4 \end{bmatrix}\end{array} \) 3) \(\begin{array}{l}\begin{bmatrix} -1 &3... View Article

If A = [latex]\begin{bmatrix} 2 &-1 \\ 1&2 \end{bmatrix}[/latex] then A² + 2A – 3I =

1) \(\begin{array}{l}\begin{bmatrix} 4 &-6 \\ 6&4 \end{bmatrix}\end{array} \) 2) 0 3) \(\begin{array}{l}\begin{bmatrix} -6 &2 \\ 2&6 \end{bmatrix}\end{array} \) 4) 5I... View Article

If A and B are square matrices of equal degree, then which one is correct among the following?

1) A + B = B + A 2) A + B = A – B 3) A – B... View Article

If A = [latex]begin{bmatrix} 3 &1 \ -1&2 end{bmatrix}[/latex], then

1) A2 + 7A – 5I = 0 2) A2 – 7A + 5I = 0 3) A2 + 5A... View Article

Let a, b, c be the real numbers. Then the following system of equations in x, y and z (x² / a²) + (y² / b²) – (z² / c²) = 1, (x² / a²) – (y² / b²) + (z² / c²) = 1,(- x² / a²) + (y² / b²) + (z² / c²) = 1 has

1) no solution 2) unique solution 3) infinitely many solutions 4) finitely many solutions Solution: (4) finitely many solutions \(\begin{array}{l}x=\delta_{1}/\delta,y=\delta_{2}/\delta,c=\delta_{3}/\delta\\... View Article

If AB = A and BA = B, then B² is equal to

1) B 2) A 3) – B 4) B3 Solution: (1) B AB = A and BA = B BA... View Article

If n ≥ 2 is an integer A = [latex]\begin{bmatrix} cos(2\pi/n) &sin(2\pi/n) &0 \\ -sin(2\pi/n)&cos(2\pi/n) &0 \\ 0&0 &1 \end{bmatrix}[/latex] and I is the identity matrix of order 3. Then

1) An = I and An-1 ≠ 1 2) Am ≠ I for any positive integer m 3) A is... View Article

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