Let A be a symmetric matrix of order 2 with integer entries. If sum of diagonal elements of A ^{2} i

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3 and 4 .Determinants and Matrices

medium

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A ^{2}$ is $1,$ then the possible number of such matrices is

$4$

$1$

$6$

$12$

(JEE MAIN-2021)

Solution

$A=\left(\begin{array}{ll}a & b \\ b & c\end{array}\right), \quad a, b, c \in I$

$A^{2}=\left(\begin{array}{ll}a & b \\ b & c\end{array}\right)\left(\begin{array}{ll}a & b \\ b & c\end{array}\right)=\left(\begin{array}{cc}a^{2}+b^{2} & b(a+c) \\ b(a+c) & b^{2}+c^{2}\end{array}\right)$

Sum of the diagonal entries of $A^{2}=a^{2}+2 b^{2}+c^{2}$

Given $a^{2}+2 b^{2}+c^{2}=1, a, b, c \in I$

$b =0$ and $a ^{2}+ c ^{2}=1$

Case $-1: a=0 \Rightarrow c=\pm 1 \quad$ $(2-$matrices)

Case $- 2 : c=0 \Rightarrow a=\pm 1 \quad$ $(2-$matrices)

Total $=4$ matrices

Standard 12

Mathematics

Show that the matrix $B ^{\prime}A B$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric.

medium

If $A$ and $B$ are symmetric matrices of the same order, then show that $AB$ is symmetric if and only if $A$ and $B$ commute, that is $AB = BA$.

medium

If the matrix $\mathrm{A}$ is both symmetric and skew symmetric, then

easy

Let $A=\left[\begin{array}{lll}x & y & z \\ y & z & x \\ z & x & y\end{array}\right], \quad$ where $x, y$ and $z$ are real numbers such that $x + y + z >0$ and $xyz =2$ If $A ^{2}= I _{3},$ then the value of $x ^{3}+ y ^{3}+ z ^{3}$ is …………

hard

(JEE MAIN-2021)

The set of all values of $t \in R$, for which the matrix

$\left[\begin{array}{ccc}e^t & e^{-t}(\sin t-2 \cos t) & e^{-t}(-2 \sin t-\cos t) \\e^t & e^{-t}(2 \sin t+\cos t) & e^{-t}(\sin t-2 \cos t) \\e^t & e^{-t} \cos t & e^{-t} \sin t \end{array}\right]$ is invertible.

hard

(JEE MAIN-2023)

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A ^{2}$ is $1,$ then the possible number of such matrices is

Solution

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The set of all values of $t \in R$, for which the matrix

$\left[\begin{array}{ccc}e^t & e^{-t}(\sin t-2 \cos t) & e^{-t}(-2 \sin t-\cos t) \\e^t & e^{-t}(2 \sin t+\cos t) & e^{-t}(\sin t-2 \cos t) \\e^t & e^{-t} \cos t & e^{-t} \sin t \end{array}\right]$ is invertible.