Let M be a 3 x 3 matrix such that , and . If , then x + y + z equals : [2026]

Question

Let M be a 3 x 3 matrix such that $M = (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}) = (\begin{matrix} 1 \\ 2 \\ 3 \end{matrix})$ , $M = (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ 1 \\ 2 \end{matrix})$ and $M = (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}) = (\begin{matrix} - 1 \\ 1 \\ 1 \end{matrix})$ .

If $M = (\begin{matrix} x \\ y \\ z \end{matrix}) = (\begin{matrix} 1 \\ 7 \\ 11 \end{matrix})$ , then x + y + z equals : [2026]

Smart Achievers · Accepted Answer

(2)

Let $M = [\begin{matrix} a & b & c \\ p & q & r \\ l & m & n \end{matrix}]$

Given, $M = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] \Rightarrow [\begin{matrix} a & b & c \\ p & q & r \\ l & m & n \end{matrix}] [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

$\Rightarrow$ a = 1, p = 2, $l$ = 3

Also, $[\begin{matrix} a & b & c \\ p & q & r \\ l & m & n \end{matrix}] [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} 0 \\ 1 \\ 2 \end{matrix}]$

$\Rightarrow$ b = 0, q = 1, m = 2

And $[\begin{matrix} a & b & c \\ p & q & r \\ l & m & n \end{matrix}] [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}] = [\begin{matrix} - 1 \\ 1 \\ 1 \end{matrix}]$

$\Rightarrow$ c = –1, r = 1, n = 1

$∴ M = [\begin{matrix} 1 & 0 & - 1 \\ 2 & 1 & 1 \\ 3 & 2 & 1 \end{matrix}]$

Now, $[\begin{matrix} 1 & 0 & - 1 \\ 2 & 1 & 1 \\ 3 & 2 & 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 7 \\ 11 \end{matrix}]$

$\Rightarrow$ x – z = 1 ... (i)

2x + y + z = 7 ... (ii)

3x + 2y + z = 11 ... (iii)

From (i), (ii) and (iii), we get

x = 2, y = 2, z = 1

$∴$ x + y + z = 5

Solution

1 4

2 5

3 7

4 11

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Solution

Q. Let M be a 3 x 3 matrix such that M=(100)=(123), M=(010)=(012) and M=(001)=(–111). If M=(xyz)=(1711), then x + y + z equals : [2026]

1 4

2 5

3 7

4 11

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