41.

Let A be a non-singular square matrix. Then,$\left|\mathrm{adj} \mathrm{A}\right|$ is equal to

• $\left|\mathrm{A}\right|$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 1}$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 2}$

• None of these

If P is a 3 x 3 matrix such that P^T = 2P + I, where p^T is the transpose of P and I is 3 x 3 identity, then there exists a column matrix X = $\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right] \ne \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$ such that PX is equal to

• $\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

• X

• 2X

• - X

If A = $\left[\begin{array}{ccc}0& 1& - 1\\ 2& 1& 3\\ 3& 2& 1\end{array}\right]$, then (A(adj A)A^{- 1}) is equal to

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

• $\left[\begin{array}{ccc}0& 1/6& - 1/6\\ 2/6& 1/6& 3/6\\ 3/6& 2/6& 1/6\end{array}\right]$

• $\left[\begin{array}{ccc}- 6& 0& 0\\ 0& - 6& 0\\ 0& 0& - 6\end{array}\right]$

• None of these

Let A = $\left[\begin{array}{ccc}1& - 1& 1\\ 2& 1& - 3\\ 3& 2& 1\end{array}\right]$ and 10B = $\left[\begin{array}{ccc}4& 2& 2\\ - 5& 0& \mathrm{\alpha }\\ 1& - 2& 3\end{array}\right]$. If B is the inverse of A, then $\mathrm{\alpha }$ is

• 5

• - 2

• 1

• - 1

If A and B are two matrices such that rank of A = m and rank of B = n, then

• rank (AB) $\ge$ rank (B)

• rank (AB) $\ge$ rank (A)

• rank (AB) $\le$ min (rank A, rank B)

• rank(AB) = mn

If the matrix A = $\left[\begin{array}{ccc}1& 3& 1\\ - 1& 2& - 3\\ 0& 1& 2\end{array}\right]$, then adj (adj A) is equal to

• $\left[\begin{array}{ccc}12& 36& 12\\ - 12& 24& - 36\\ 0& 12& 24\end{array}\right]$

• $\left[\begin{array}{ccc}12& 26& - 12\\ 24& 36& - 36\\ 0& 12& - 24\end{array}\right]$

• $\left[\begin{array}{ccc}12& - 12& 36\\ 24& - 24& - 36\\ 0& 12& 24\end{array}\right]$

• None of the above

If A = $\left[\begin{array}{ccc}1& 3& 1\\ 2& 1& - 1\\ 3& 0& 1\end{array}\right]$ then rank (A) is equal to

• 4

• 1

• 2

• 3

Find the value of k for which the\ simultaneous equations x + y + z = 3; x + 2 y + 3Z = 4 and x + 4 y + kz = 6 will not have a unique solution.

• 0

• 5

• 6

• 7

If A = $\left[\begin{array}{cc}3& 4\\ 5& 7\end{array}\right]$ then A . (adj A) is equal to

• A

• $\left|\mathrm{A}\right|$

• $\left|\mathrm{A}\right| . \mathrm{I}$

• None of these

If the points (x₁, y₁), (x₂, y₂) and (x₃, y₃) are collinear, then the rank of the matrix $\left[\begin{array}{ccc}{\mathrm{x}}_1& {\mathrm{y}}_1& 1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& 1\\ {\mathrm{x}}_3& {\mathrm{y}}_3& 1\end{array}\right]$ will always be less than

• 3

• 2

• 1

• None of these