JEE MATHEMATICS MCQ: MATRICES

1 Easy

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Understanding +4 | -1

Let

A

be a

3 \times 3

matrix and

S=\left\{\left(\begin{array}{l} x \\ y \\ z \end{array}\right) x, y, z \in {R}\right\}

Define

f: S \rightarrow S

f\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=A\left(\begin{array}{l} x \\ y \\ z \end{array}\right)

Suppose

f\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right) \Rightarrow x=y=z=0

. Then

View answer and explanation

Correct option: A

f

is one-to-one

Option A (

f

is one-to-one) is correct because the condition

f(\begin{array}{l}x \\ y \\ z\end{array})=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right) \Rightarrow x=y=z=0

implies that the null space of A contains only the zero vector, meaning that the linear transformation is injective (one-to-one).

More Questions from MATRICES

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\left|\begin{array}{ccc}a & b-c & b+c \\ c+a & b & c-a \\ a-b & a+b & c\end{array}\right|=0

a, b, c

a x+b y+c=0

{x_2} - {x_3} = 1,\; - {x_1} + 2{x_3} = - 2,\;{x_1} - 2,\;{x_1} - 2{x_2} = 3

2x + 3y + 1 = 0,3x + y - 2 = 0

ax + 2y - b = 0

\left|\begin{array}{ccc}1 & \left(\alpha^{2 x}-\alpha^{-2 x}\right)^2 & \left(\alpha^{2 x}+\alpha^{-2 x}\right)^2 \\ 1 & \left(\beta^{2 x}-\beta^{-2 x}\right)^2 & \left(\beta^{2 x}+\beta^{-2 x}\right)^2 \\ 1 & \left(\gamma^{2 x}-\gamma^{-2 x}\right)^2 & \left(\gamma^{2 x}+\gamma^{-2 x}\right)^2\end{array}\right|

\omega

\left| {\begin{array}{cccccccccccccccccccc} 1&\omega &{{\omega ^2}}\\ \omega &{{\omega ^2}}&1\\ {{\omega ^2}}&1&\omega \end{array}} \right|

A

2 \times 2

A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]

|A|