If a_ij = \({1\over2}(3i-2j)\) and A = [a_ij]_2x2 is

What must be the matrix X, if 2x +\(\begin{bmatrix} 1& 2 \\ 3 & 4 \end{bmatrix}=\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}?\)

The value of the determinant of A = \(\begin{bmatrix} 0&a &-b \\ -a & 0 & c \\ b & -c & 0 \end{bmatrix}is\)

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in geometric progression with the same common ratio, then the points (x₁, y₁ ), (x₂, y₂), (x₃, y₃ ) are

If a \(\neq\) b, b, c satisfy \(\begin{vmatrix} a&2b &2c \\3 & b & c \\ 4 & a & b \end{vmatrix}=0,\) then abc =

The value of \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}\) is

If \(\overrightarrow{a}+2\overrightarrow{b}\) and \(3\overrightarrow{a}+m\overrightarrow{b}\) are parallel, then the value of m is

If \(|\overrightarrow{a}|=13,|\overrightarrow{b}|=5\)  and \(\overrightarrow{a}.\overrightarrow{b}=60^o\) then \(|\overrightarrow{a}\times\overrightarrow{b}|\) is

Vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are inclined at an angle \(\theta =120^o\). If \(|\overrightarrow{a}|=1,|\overrightarrow{b}|=2,\) then \([(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})]^2\) is equal to

If \(\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \({\pi\over 6},\) then the area of the triangle formed by these two vectors as two sides, is

Simplify : \(sec \theta \begin{bmatrix} sec \theta & tan\theta \\ tan\theta & sec\theta \end{bmatrix}\)-\(tan \theta \begin{bmatrix} tan \theta & sec\theta \\ sec\theta & tan\theta \end{bmatrix}\)

If A =\(\begin{bmatrix} 0 &c &b \\ c & 0 &a \\ b & a & 0 \end{bmatrix}\), compute A²

Consider the matrix Aa=\(\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\)
Find all possible real values of α satisfying the condition \(A\alpha +A^T_{\alpha}=I\)

If A = \(\begin{bmatrix} 4 & 2 \\ -1 & x \end{bmatrix}\) and such that (A - 2I)(A - 3I) = O, find the value of x.

Can a vector have direction angles 30°, 45°, 60°?

Find the direction cosines of \(\overrightarrow{AB},\) where A is (2, 3, 1) and B is (3, - 1, 2).

Find the direction cosines of the line joining (2, 3, 1) and (3, - 1, 2).

If \(\overrightarrow{a}\) and \(\overrightarrow{b}\)are two vectors such that | \(\overrightarrow{a}\) | = 10, | \(\overrightarrow{b}\) | = 15 and \(\overrightarrow{a}\).\(\overrightarrow{b}\) = 75 \(\sqrt{2}\), find the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\).

Find the angle between the vectors  \(2\hat{i}+3\hat{j}-6\hat{k}\) and \(6\hat{i}-3\hat{j}+2\hat{k}\)

Five mangoes and 4 apples are in a box. If two fruits are chosen at random, find the probability that (i) one is a mango and the other is an apple (ii) both are of the same variety.

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B= \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\)
verify (AB)^T = B^TA^T

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\) verify (A+B)^T = A^T + B^T

If A =\(\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}\) verify (A - B)^T = A^T - B^T

If A_i, B_i, C_i are the cofactors of a_i, b_i,c_i, respectively, i = 1 to 3 in
|A| = \(\begin{vmatrix} a_1 &b_1 &c_1 \\ a_2 & b_2 &c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}\), show that \(\begin{vmatrix} A_1 &B_1 &C_1 \\ A_2 & B_2 &C_2 \\ A_3 & B_3 & C_3 \end{vmatrix}\) = |A|²

Identify the singular and non-singular matrices:\(\begin{bmatrix} 1&2 &3 \\ 4 & 5 &6 \\ 7 & 8 & 9 \end{bmatrix}\)

If \(\begin{bmatrix} 0 & p& 3 \\ 2 & q^2 & -1 \\ r & 1 & 0 \end{bmatrix}\) is skew-symmetric, find the values of p, q, and r.

Construct the matrix \(A=[a_{ij}]_{3\times 3}\), where \(a_{ij}=i-j.\) State whether A is symmetric or skew-symmetric.

Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.