Let $A$ be a $3 \times 3$ real matrix such that $A\left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ 0 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ 0 \end{array}} \right);A\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 1} \\ 0 \\ 1 \end{array}} \right)\,\,{\text{and}}\,\,A\left( {\begin{array}{*{20}{c}} 0 \\ 0 \\ 1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ 2 \end{array}} \right).$ If $X = {({x_1},{x_2},{x_3})^T}$ and $I$ is an identity matrix of order 3, then the system $(A - 2I)X = \left( {\begin{array}{*{20}{c}} 4 \\ 1 \\ 1 \end{array}} \right)$ has:

JEE MATHEMATICS MCQ: MATRICES

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Analyzing +4 | -1

Let

A

be a

3 \times 3

real matrix such that

A\left( {\begin{array}{cccccccccccccccccccc} 1 \\ 1 \\ 0 \end{array}} \right) = \left( {\begin{array}{cccccccccccccccccccc} 1 \\ 1 \\ 0 \end{array}} \right);A\left( {\begin{array}{cccccccccccccccccccc} 1 \\ 0 \\ 1 \end{array}} \right) = \left( {\begin{array}{cccccccccccccccccccc} { - 1} \\ 0 \\ 1 \end{array}} \right)\,\,{\text{and}}\,\,A\left( {\begin{array}{cccccccccccccccccccc} 0 \\ 0 \\ 1 \end{array}} \right) = \left( {\begin{array}{cccccccccccccccccccc} 1 \\ 1 \\ 2 \end{array}} \right).

X = {({x_1},{x_2},{x_3})^T}

and

I

is an identity matrix of order 3, then the system

(A - 2I)X = \left( {\begin{array}{cccccccccccccccccccc} 4 \\ 1 \\ 1 \end{array}} \right)

has:

View answer and explanation

Correct option: B

Infinite many solutions

Option B (Infinite many solutions) is correct because the system of equations derived from

(A - 2I)X = \begin{pmatrix} 4 \\ 1 \\ 1 \end{pmatrix}

has dependent equations, leading to infinite solutions.