Let A be a symmetric matrix of order 2 with i | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $A = \begin{bmatrix} a & b \ b & c \end{bmatrix}$,where $a, b, c \in \mathbb{Z}$.Then $A^{2} = \begin{bmatrix} a & b \ b & c \end{bmatrix} \

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(A) Let $A = \begin{bmatrix} a & b \ b & c \end{bmatrix}$,where $a, b, c \in \mathbb{Z}$.Then $A^{2} = \begin{bmatrix} a & b \ b & c \end{bmatrix} \begin{bmatrix} a & b \ b & c \end{bmatrix} = \begin{bmatrix} a^{2} + b^{2} & ab + bc \ ab + bc & b^{2} + c^{2} \end{bmatrix}$.The sum of the diagonal elements of $A^{2}$ is given by $	ext{tr}(A^{2}) = a^{2} + b^{2} + b^{2} + c^{2} = a^{2} + 2b^{2} + c^{2}$.We are given that $a^{2} + 2b^{2} + c^{2} = 1$,where $a, b, c \in \mathbb{Z}$.Since $a^{2}, b^{2}, c^{2} \ge 0$,for the sum to be $1$,we must have $b^{2} = 0$,which implies $b = 0$.Then the equation reduces to $a^{2} + c^{2} = 1$.Since $a, c \in \mathbb{Z}$,the possible integer solutions for $(a, c)$ are:$1$. If $a = 0$,then $c^{2} = 1 \Rightarrow c = \pm 1$. This gives two matrices: $\begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix}$ and $\begin{bmatrix} 0 & 0 \ 0 & -1 \end{bmatrix}$.$2$. If $c = 0$,then $a^{2} = 1 \Rightarrow a = \pm 1$. This gives two matrices: $\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}$ and $\begin{bmatrix} -1 & 0 \ 0 & 0 \end{bmatrix}$.Thus,the total number of such matrices is $2 + 2 = 4$.

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

Similar Questions

If $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$,then $A^5 = $ (in $A$)

$A = [a_{ij}]_{m \times n}$ is a square matrix,if

If $A = \begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix}$ and $I$ is the identity matrix of order $2$,then $(A - 2I)(A - 3I) = $

Let $A = \begin{bmatrix} 0 & 1 \\ 1 & k \end{bmatrix}$, $k \in R$ and $A^3 = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. If $d = 228$, then $b + c =$

If $\begin{bmatrix} x & 0 \\ 1 & y \end{bmatrix} + \begin{bmatrix} -2 & 1 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 5 \\ 6 & 3 \end{bmatrix} - \begin{bmatrix} 2 & 4 \\ 2 & 1 \end{bmatrix}$,then find the values of $x$ and $y$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module