Suppose A=left[begin{array}{ll}a & b c & dend{array}right] is a real matrix with non-zero en

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3 and 4 .Determinants and Matrices

hard

Suppose $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is a real matrix with non-zero entries, $\alpha d-b c=0$ and $A^2=A$. Then, $a + d$ equals

$1$

$2$

$3$

$4$

(KVPY-2018)

Solution

(a)

We have

$\begin{aligned} A &=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \\ A^2 &=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \\ A^2 &=\left[\begin{array}{ll}a^2+b c & a b+b d \\ a c+c d & b c+d^2\end{array}\right] \end{aligned}$

Given, $A^2=A$ and $a d-b c=0$

$\therefore\left[\begin{array}{ll} a^2+b c & a b+b d \\ a c+c d & b c+d^2 \end{array}\right]=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$

$a b+b d =b$

$b(a+d) =b$

$a+d =1$

Standard 12

Mathematics

If $A = \left[ {\begin{array}{*{20}{c}}
1&0\\
{\frac{1}{2}}&1
\end{array}} \right]$ , then $A^{50}$ is

normal

If $A = dig(2, – 1,\,3),B = dig( – 1,\,3,\,2)$, then ${A^2}B = $

medium

Which of the given values of $x$ and $y$ make the following pair of matrices equal

$\left[\begin{array}{cc}3 x+7 & 5 \\ y+1 & 2-3 x\end{array}\right]=\left[\begin{array}{cc}0 & y-2 \\ 8 & 4\end{array}\right]$

easy

If $A = \left[ {\begin{array}{*{20}{c}}2&2\\a&b\end{array}} \right]$ and ${A^2} = O$, then $(a,b) = $

easy

If $B = \left[ {\begin{array}{*{20}{c}}
5&{2\alpha }&1\\
0&2&1\\
\alpha &3&{ – 1}
\end{array}} \right]$ is the inverse of a $3 \times 3$ matrix $A$, then the sum of all values of $\alpha $ for which $det\, (A) + 1 = 0$, is

hard

(JEE MAIN-2019)

Suppose $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is a real matrix with non-zero entries, $\alpha d-b c=0$ and $A^2=A$. Then, $a + d$ equals

Solution

Similar Questions

If $A = \left[ {\begin{array}{*{20}{c}} 1&0\\ {\frac{1}{2}}&1 \end{array}} \right]$ , then $A^{50}$ is

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If $A = \left[ {\begin{array}{*{20}{c}}2&2\\a&b\end{array}} \right]$ and ${A^2} = O$, then $(a,b) = $

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If $A = \left[ {\begin{array}{*{20}{c}}
1&0\\
{\frac{1}{2}}&1
\end{array}} \right]$ , then $A^{50}$ is

Which of the given values of $x$ and $y$ make the following pair of matrices equal

$\left[\begin{array}{cc}3 x+7 & 5 \\ y+1 & 2-3 x\end{array}\right]=\left[\begin{array}{cc}0 & y-2 \\ 8 & 4\end{array}\right]$

If $B = \left[ {\begin{array}{*{20}{c}}
5&{2\alpha }&1\\
0&2&1\\
\alpha &3&{ – 1}
\end{array}} \right]$ is the inverse of a $3 \times 3$ matrix $A$, then the sum of all values of $\alpha $ for which $det\, (A) + 1 = 0$, is