New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Matrices and Determinants question

2003 · Shift 0 · Q79

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Mathematics
Matrices and Determinants
2003 · Shift 0 · Q79

Matrices and Determinants question

2003 · Shift 0 · Q79

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1

If

A = \left[ {\begin{matrix} a & b \\ b & a \\ \end{matrix} } \right]

and

{A^2} = \left[ {\begin{matrix} \alpha & \beta \\ \beta & \alpha \\ \end{matrix} } \right]

, then

A
$\alpha = 2ab,\,\beta = {a^2} + {b^2}$
B
$\alpha = {a^2} + {b^2},\,\beta = ab$
C
$\alpha = {a^2} + {b^2},\,\beta = 2ab$
D
$\alpha = {a^2} + {b^2},\,\beta = {a^2} - {b^2}$

View written solutionFree

Correct answer: C

We are given $A=\begin{bmatrix}a&b\\ b&a\end{bmatrix}$ and $A^2=\begin{bmatrix}\alpha&\beta\\ \beta&\alpha\end{bmatrix}.$
Compute $A^2$ by matrix multiplication:

\begin{bmatrix}a&b\\ b&a\end{bmatrix} \begin{bmatrix}a&b\\ b&a\end{bmatrix}.$$ 3. Multiply entry-wise: - First row, first column: $$a\cdot a+b\cdot b=a^2+b^2$$ - First row, second column: $$a\cdot b+b\cdot a=ab+ab=2ab$$ - Second row, first column: $$b\cdot a+a\cdot b=ba+ab=2ab$$ - Second row, second column: $$b\cdot b+a\cdot a=b^2+a^2=a^2+b^2$$ So, $$A^2=\begin{bmatrix}a^2+b^2&2ab\\ 2ab&a^2+b^2\end{bmatrix}.$$ 4. Comparing with $$A^2=\begin{bmatrix}\alpha&\beta\\ \beta&\alpha\end{bmatrix},$$ we get $$\alpha=a^2+b^2,\qquad \beta=2ab.$$ 5. Hence the correct option is: $$\boxed{\text{C}}$$

More from Matrices and Determinants

If $1, ω, ω^{2}$ are the cube roots of unity, then $Δ = 1 ω^{n} ω^{2 n} ω^{n} ω^{2 n} 1 ω^{2 n} 1 ω^{n}$ …2003 · MCQ
If the system of linear equations $x + 2 a y + a z = 0; x + 3 b y + b z = 0; x + 4 cy + cz = 0;$ has a non - zero solution, then $a, b, c$ .2003 · MCQ
If $a > 0$ and discriminant of $a x^{2} + 2 b x + c$ is $- v e$ , then $a b a x + b b c b x + c a x + b b x + c 0$ is equal to2002 · MCQ
Let $A = [α 6 - 1 β], α > 0$ , such that $det (A) = 0$ and $α + β = 1$ . If I denotes $2 \times 2$ identity matrix, then the matrix…2025 · MCQ
Let $a \in R$ and $A$ be a matrix of order $3 \times 3$ such that $det (A) = - 4$ and $A + I = 12 a a 11 102$ , where $I$ is the identity matrix…2025 · MCQ
If the system of linear equations $3 x + y + β z = 3 2 x + α y - z = - 3 x + 2 y + z = 4$ has infinitely many solutions, then the value of $22 β - 9 α$ is :2025 · MCQ
If the system of equations $2 x + λ y + 3 z = 5 3 x + 2 y - z = 7 4 x + 5 y + μ z = 9$ has infinitely many solutions, then $(λ^{2} + μ^{2})$ is equal to :2025 · MCQ
Let $A$ be a $3 \times 3$ real matrix such that $A^{2} (A - 2 I) - 4 (A - I) = O$ , where $I$ and $O$ are the identity and null matrices, respectively. If $A^{5} = α A^{2} + β A + γ I$ , where $γ$ , and $γ$ are real constants, then $α + β + γ$ …2025 · MCQ