Let M be a 3 - 3 matrix satisying - M bmatrix 0 11 W0 e n d b m a t r i x - beginbmatrix -1 12113 e n d b m a t r i x - MWeginbmatrix 111-1110lendbmatrix - bmatrix 111111-1lendbmatrix - and - M bmatrix 1111111lendbmatrix - begin bmatrix 0 10 112lendbmatrix - Then- the sum of the diagonal entries of M is

Let M = [ a_1 amp; a_2 amp;a_3; b_1 amp; b_2 amp; b_3; c_1 amp; c_2 amp; c_3 ] ∴ M [ 0; 1; 0 ] = [ 1; 2; 3 ] M [ 1; 1; 0 ]=[ 1; 1; 1 ] ...

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Standard IX

Mathematics

Equality of Matrices

Let M be a 3 ...

Question

Let M be a $3 \times 3$ matrix satisying $M ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} - 1 23 \end{matrix} ⎤ ⎥ ⎦, M ⎡ ⎢ ⎣ \begin{matrix} 1 - 1 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 11 - 1 \end{matrix} ⎤ ⎥ ⎦, a n d M ⎡ ⎢ ⎣ \begin{matrix} 111 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 0012 \end{matrix} ⎤ ⎥ ⎦ .$
Then, the sum of the diagonal entries of M is ___

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Solution

Let $M = ⎡ ⎢ ⎣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ⎤ ⎥ ⎦ ∴ M ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} - 1 23 \end{matrix} ⎤ ⎥ ⎦ M ⎡ ⎢ ⎣ \begin{matrix} 1 - 1 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 11 - 1 \end{matrix} ⎤ ⎥ ⎦$
$M ⎡ ⎢ ⎣ \begin{matrix} 111 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 0012 \end{matrix} ⎤ ⎥ ⎦ \Rightarrow ⎡ ⎢ ⎣ \begin{matrix} a_{2} b_{2} c_{2} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} - 1 23 \end{matrix} ⎤ ⎥ ⎦, ⎡ ⎢ ⎣ \begin{matrix} a_{1} - a_{2} b_{1} - b_{2} c_{1} - c_{2} \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 11 - 1 \end{matrix} ⎤ ⎥ ⎦, ⎡ ⎢ ⎣ \begin{matrix} a_{1} + a_{2} + a_{3} b_{1} + b_{2} + b_{3} c_{1} + c_{2} + c_{3} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 0012 \end{matrix} ⎤ ⎥ ⎦ \Rightarrow a_{2} = - 1, b_{2} = 2, c_{2} = 3, a_{1} - a_{2} = 1, b_{1} - b_{2} = 1, c_{1} - c_{2} = - 1 \Rightarrow a_{1} + a_{2} + a_{3} = 0, b_{1} + b_{2} + b_{3} = 0 c_{1} + c_{2} + c_{3} = 12 ∴ a_{1} = 0, b_{2} = 2, c_{3} = 7$
$\Rightarrow$ Sum of diagonal elements = 0 + 2 + 7 = 9.

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Similar questions

Q. Let M be a

3 \times 3

matrix satisying

M ⎡ ⎢ ⎣ \begin{matrix} 010 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} - 1 23 \end{matrix} ⎤ ⎥ ⎦, M ⎡ ⎢ ⎣ \begin{matrix} 1 - 1 0 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 11 - 1 \end{matrix} ⎤ ⎥ ⎦, a n d M ⎡ ⎢ ⎣ \begin{matrix} 111 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 0012 \end{matrix} ⎤ ⎥ ⎦ .

Then, find the sum of the diagonal entries of M.

Let $A = [\begin{matrix} 2 & 1 0 & - 2 \end{matrix}], B = [\begin{matrix} 4 & 1 - 3 & - 2 \end{matrix}]$ and $C = [\begin{matrix} - 3 & 2 - 1 & 4 \end{matrix}]$ Find: $A^{2} + A C - 5 B$

Which of the following would be the Inverse of the matrix A found using Elementary Row Transformations where

A = $∣ ∣ ∣ ∣ \begin{matrix} 1 & 3 & 1 0 & 1 & 1 3 & 6 & 1 \end{matrix} ∣ ∣ ∣ ∣$

Q. Let

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 0 & 1 & 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

. Then for positive integer

n, A^{n}

Inverse of thefollowing matrix using elementary Row transformations would be

$= ∣ ∣ ∣ ∣ \begin{matrix} 1 & 3 & 1 0 & 1 & 1 3 & 6 & 1 \end{matrix} ∣ ∣ ∣ ∣$

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Let M be a 3×3 matrix satisying M⎡⎢⎣010⎤⎥⎦=⎡⎢⎣−123⎤⎥⎦,M⎡⎢⎣1−10⎤⎥⎦=⎡⎢⎣11−1⎤⎥⎦,and M⎡⎢⎣111⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦. Then, the sum of the diagonal entries of M is ___