Let M be a 3 times 3 matrix satisfying Mbegin | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $M = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$.From the first condition,$\begin{bmatrix} a & b & c \ d & e & f \ g

Vedclass · Accepted Answer

$9$

Vedclass · Answer

(A) Let $M = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$.From the first condition,$\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix} = \begin{bmatrix} b \ e \ h \end{bmatrix} = \begin{bmatrix} -1 \ 2 \ 3 \end{bmatrix}$. Thus,$b = -1, e = 2, h = 3$.Now,$M = \begin{bmatrix} a & -1 & c \ d & 2 & f \ g & 3 & i \end{bmatrix}$.From the second condition,$\begin{bmatrix} a & -1 & c \ d & 2 & f \ g & 3 & i \end{bmatrix} \begin{bmatrix} 1 \ -1 \ 0 \end{bmatrix} = \begin{bmatrix} a+1 \ d-2 \ g-3 \end{bmatrix} = \begin{bmatrix} 1 \ 1 \ -1 \end{bmatrix}$.Solving these,we get $a+1 = 1 \Rightarrow a = 0$,$d-2 = 1 \Rightarrow d = 3$,and $g-3 = -1 \Rightarrow g = 2$.Now,$M = \begin{bmatrix} 0 & -1 & c \ 3 & 2 & f \ 2 & 3 & i \end{bmatrix}$.From the third condition,$\begin{bmatrix} 0 & -1 & c \ 3 & 2 & f \ 2 & 3 & i \end{bmatrix} \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix} = \begin{bmatrix} -1+c \ 5+f \ 5+i \end{bmatrix} = \begin{bmatrix} 0 \ 0 \ 12 \end{bmatrix}$.From the third row,$5+i = 12 \Rightarrow i = 7$.The sum of the diagonal entries is $a + e + i = 0 + 2 + 7 = 9$.

Similar Questions

Let $A = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix}$ and $B = 7A^{20} - 20A^{7} + 2I$,where $I$ is an identity matrix of order $3 \times 3$. If $B = [b_{ij}]$,then $b_{13}$ is equal to $....$

If $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ and $\theta = \frac{2 \pi}{7}$, then $A^{100} = A \times A \times \dots \times A$ ($100$ times) is equal to:

If $\begin{bmatrix} x + y & 2x + z \\ x - y & 2z + w \end{bmatrix} = \begin{bmatrix} 4 & 7 \\ 0 & 10 \end{bmatrix}$,then the values of $x, y, z, w$ are

Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$,$B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$,and $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$. Find $A + B$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module