Matrices & DeterminantsmediumPYQ · JEE Main · 5 Apr 2026 · Shift 2 (Afternoon)Free

Matrices & Determinants: Let Matrix Pmatrix Pmatrix Pmatrix Pmatrix Pmatrix Pmatrix (JEE Main 2026)

JEE Maths question with a full step-by-step solution.

Question

Let

M

be a

3\times3

matrix such that

M\begin{pmatrix}1\\0\\0\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}

M\begin{pmatrix}0\\1\\0\end{pmatrix}=\begin{pmatrix}0\\1\\2\end{pmatrix}

, and

M\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}-1\\1\\1\end{pmatrix}

. If

M\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\7\\11\end{pmatrix}

, then

x+y+z

equals:

4

5

correct

7

11

Solution

Step 1: Read off the columns of

M

Each condition

M\vec{e}_j = \vec{v}_j

gives column

j

M

directly:

M = \begin{pmatrix}1&0&-1\\2&1&1\\3&2&1\end{pmatrix}

Step 2: Set up the linear system

\begin{cases} x - z = 1 \quad &\cdots(1) \\ 2x + y + z = 7 \quad &\cdots(2) \\ 3x + 2y + z = 11 \quad &\cdots(3) \end{cases}

Step 3: Solve the system

(3)-(2)

x+y=4 \quad\cdots(4)

From (1):

z = x-1 \quad\cdots(5)

Substituting (5) into (2):

2x+y+(x-1)=7 \Rightarrow 3x+y=8 \quad\cdots(6)

(6)-(4)

2x=4 \Rightarrow x=2

. Then

y=2

from (4) and

z=1

from (5). Step 4: Compute the sum

x+y+z = 2+2+1 = 5

Answer: (2)

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