JEE MAIN Matrices Previous Year Questions (PYQs) – Page 14 of 15
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Let $A$ be a $3 \times 3$ matrix such that $A^{2} - 5A + 7I = 0$.
\textbf{Statement I:}
$A^{-1} = \dfrac{1}{7}(5I - A)$.
\textbf{Statement II:}
The polynomial $A^{3} - 2A^{2} - 3A + I$ can be reduced to $5(A - 4I)$.
Then:
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Solution
If
$A=\begin{bmatrix}
e^{t} & e^{-t}\cos t & e^{-t}\sin t\\[4pt]
e^{t} & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t\\[4pt]
e^{t} & 2e^{-t}\sin t & -2e^{-t}\cos t
\end{bmatrix}$,
then $A$ is:
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Solution
The number of $\theta \in(0,4 \pi)$ for which the system of linear equations
$\begin{aligned}&3(\sin 3 \theta) x-y+z=2 \\\\&3(\cos 2 \theta) x+4 y+3 z=3 \\\\&6 x+7 y+7 z=9\end{aligned}$
has no solution, is :
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Solution
The values of $\lambda$ and $\mu$ such that the system of equations $x + y + z = 6$, $3x + 5y + 5z = 26$, $x + 2y + \lambda z = \mu $ has no solution, are :
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Solution
Let A = [aij] be a real matrix of order 3 $\times$ 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A3 is equal to :
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Solution
Let the system of equations
$x+2y+3z=5,\quad 2x+3y+z=9,\quad 4x+3y+\lambda z=\mu$
have infinite number of solutions. Then $\lambda+2\mu$ is equal to:
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Solution
$ \textbf{Q:}$ For the system of linear equations $x + y + z = 6,\ \alpha x + \beta y + 7z = 3,\ x + 2y + 3z = 14$, which of the following is **NOT true**?
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Solution
If the system of linear equations :
$\begin{aligned} & x+y+2 z=6 \\ & 2 x+3 y+\mathrm{az}=\mathrm{a}+1 \\ & -x-3 y+\mathrm{b} z=2 \mathrm{~b} \end{aligned}$
where $a, b \in \mathbf{R}$, has infinitely many solutions, then $7 a+3 b$ is equal to :
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Solution
If the system of linear equations
$x-4y+7z=g$,
$3y-5z=h$,
$-2x+5y-9z=k$
is consistent, then:
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Solution
Let $A=\begin{pmatrix}1&0&0\\[2pt]0&4&-1\\[2pt]0&12&-3\end{pmatrix}$. Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to:
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Solution
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