Let M be a 3 × 3 matrix satisfying M [ 0 1 0 ] = [ - 1 2 3 ] , M [ 1 - 1 0 ] = [ 1 1 - 1 ] , and M [ 1 1 1 ] = [ 0 0 12 ] . Then the sum of the diagonal entries of M is [2011]

Question

Let $M$ be a $3 \times 3$ matrix satisfying

$M [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} - 1 \\ 2 \\ 3 \end{matrix}],$ $M [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}],$ and $M [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 12 \end{matrix}] .$ Then the sum of the diagonal entries of $M$ is [2011]

Smart Achievers · Accepted Answer

(9)

Let $3 \times 3$ then $[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}] [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} - 1 \\ 2 \\ 3 \end{matrix}]$

$\Rightarrow b_{1} = - 1,$ $b_{2} = 2,$ $b_{3} = 3$

$[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}] [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}]$

$\Rightarrow a_{1} - b_{1} = 1,$ $a_{2} - b_{2} = 1,$ $a_{3} - b_{3} = - 1$

$\Rightarrow a_{1} = 0,$ $a_{2} = 3,$ $a_{3} = 2$

and $[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}] [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 12 \end{matrix}]$

$\Rightarrow a_{3} + b_{3} + c_{3} = 12$ $\Rightarrow c_{3} = 7$

$∴$ $Sum of diagonal elements = a_{1} + b_{2} + c_{3} = 0 + 2 + 7 = 9$

Solution

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Solution

Q. Let M be a 3×3 matrix satisfying M[010]=[-123], M[1-10]=[11-1], and M[111]=[0012]. Then the sum of the diagonal entries of M is [2011]

Ans. (9) Let M=[a1b1c1a2b2c2a3b3c3], then [a1b1c1a2b2c2a3b3c3][010]=[-123] ⇒b1=-1, b2=2, b3=3 [a1b1c1a2b2c2a3b3c3][1-10]=[11-1] ⇒a1-b1=1, a2-b2=1, a3-b3=-1 ⇒a1=0, a2=3, a3=2 and [a1b1c1a2b2c2a3b3c3][111]=[0012] ⇒a3+b3+c3=12⇒c3=7 ∴ Sum of diagonal elements=a1+b2+c3=0+2+7=9

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