If $A$ is a non-singular matrix of order $3$ such that $(A-2I)(A-4I)=0$,then $\frac{1}{6}A + \frac{4}{3}A^{-1}$ is (where $I$ is a unit matrix of order $3$ and $0$ is a null matrix of order $3$).
- A$6I$
- B$I$
- C$0$
- D$2I$
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If $A$ is a square matrix and $A^2+I=2 A$, then $A^9=$
DifficultTS EAMCET 2024
View Solution$A$ and $B$ are two square matrices such that $A^2B = BA$. If $(AB)^{10} = A^K B^{10}$,then $k$ is:
Let $M = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $N = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$. Then $N M^{10} N^{-1} =$
Consider the following linear equations:
$ax+by+cz=0$,$bx+cy+az=0$,$cx+ay+bz=0$
Match the conditions/expressions in Column $I$ with statements in Column $II$:
Column $I$ Column $II$ $(A)$ $a+b+c \neq 0$ and $a^2+b^2+c^2=ab+bc+ca$ $(p)$ The equations represent planes meeting only at a single point. $(B)$ $a+b+c=0$ and $a^2+b^2+c^2 \neq ab+bc+ca$ $(q)$ The equations represent the line $x=y=z$. $(C)$ $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq ab+bc+ca$ $(r)$ The equations represent identical planes. $(D)$ $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$ $(s)$ The equations represent the whole of the three-dimensional space.
$ax+by+cz=0$,$bx+cy+az=0$,$cx+ay+bz=0$
Match the conditions/expressions in Column $I$ with statements in Column $II$:
| Column $I$ | Column $II$ |
| $(A)$ $a+b+c \neq 0$ and $a^2+b^2+c^2=ab+bc+ca$ | $(p)$ The equations represent planes meeting only at a single point. |
| $(B)$ $a+b+c=0$ and $a^2+b^2+c^2 \neq ab+bc+ca$ | $(q)$ The equations represent the line $x=y=z$. |
| $(C)$ $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq ab+bc+ca$ | $(r)$ The equations represent identical planes. |
| $(D)$ $a+b+c=0$ and $a^2+b^2+c^2=ab+bc+ca$ | $(s)$ The equations represent the whole of the three-dimensional space. |
Let $p, q, r$ be three real numbers satisfying $[p \, q \, r] \begin{bmatrix} 2 & p & q \\ -3 & q & -p+r \\ 12 & r & -q+3r \end{bmatrix} = [5 \, b \, c]$. Then the minimum value of $(b+c)$ is:
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