If A = \(\begin{bmatrix}\lambda & 1 \\ -1 & -\lambda \end{bmatrix}\), then for what value of \(\lambda\), A² = O?

If A is a square matrix, then which of the following is not symmetric?

If A and B are symmetric matrices of order n, where (A \(\neq\) B), then

If the points (x,−2), (5, 2), (8, 8) are collinear, then x is equal to

If A is skew-symmetric of order n and C is a column matrix of order n \(\times\) 1, then C^T AC is

Suppose that a matrix has 12 elements. What are the possible orders it can have? What if it has 7 elements?

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

Construct an m \(\times\) n matrix A = [a_ij], where a _ij is given by
\(a_{ij}={|3i-4j|\over 4}with \ m=3,n=4\)

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

Show that the points (a, b + c), (b, c + a), and (c, a + b) are collinear

If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k.

In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section of Express the given information as a column matrix. Using sclar multiplication find the total number of each kind in all the colleges.

Solve for x if \(\left[\begin{array}{lll} x & 2 & -1 \end{array}\right]\)\(\begin{bmatrix} 1&1 &2 \\ -1 & -4 &1 \\ -1 &-1 &-2 \end{bmatrix}\)\(\begin{bmatrix} x \\ 2 \\ 1 \end{bmatrix}\)=0

A fruit shop keeper prepares 3 different varieties of gift packages. Pack-I contains 6 apples, 3 oranges, and 3 pomegranates. Pack-II contains 5 apples, 4 oranges and 4 pomegranates and Pack –III contains 6 apples, 6 oranges and 6 pomegranates. The cost of an apple, an orange and a pomegranate respectively are Rs. 30, Rs. 15 and Rs. 45. What is the cost of preparing each package of fruits?

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^T = \(\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}\) and B =  \(\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}\), verify (A + B)^T = A^T + B^T = B^T + A^T

Identify the singular and non-singular matrices:\(\begin{bmatrix} 0&a-b &k \\ b-a & 0 &5 \\ -k & -5 & 0 \end{bmatrix}\)

If A = \(\left[ \begin{matrix} \alpha & 0 \\ 1 & 1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \end{matrix} \right] \) find the values of \(\alpha\) for which A² = B.

If A = \(\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y \end{bmatrix}\) is a matrix such that AA^T = 9I, find the values of x and y.

If \(\lambda =-2\) , determine the value of \(\begin{vmatrix} 0& 2\lambda &1 \\ \lambda^2 &0 &3\lambda^3+1 \\ -1 &6\lambda-1 &0 \end{vmatrix}\) .