Matrices and Determinants PYQs - Last 10 Years

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Practice 96 AP EAPCET Mathematics questions from Matrices and Determinants. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.

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Last 10 Years Matrices and Determinants Questions

Showing 46 of 96 filtered questions.

1Matrices And Determinants

$\left|\begin{array}{ccc}a & b & c \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1\end{array}\right|$ is not equal to

2Matrices And Determinants

If $A=\left[\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right]$ and $\alpha A^2+\beta A=2 I$ for some $\alpha, \beta \in R$, then $\alpha+\beta=$

3Matrices And Determinants

$$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{array}\right|= $$

4Matrices And Determinants

The system of equations

\(x+2 y+3 z=6, x+3 y+5 z=9 \text {, }\)

$2 x+5 y+a z=12$ has no solution when $a=$

5Matrices And Determinants

While solving a system of linear equations $A X=B$ using Cramer's rule with the usual notation if$$ \Delta=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 2 & -1 & 2 \\ -1 & 1 & 5 \end{array}\right|, \Delta_1=\left|\begin{array}{ccc} 5 & 1 & 1 \\ 4 &...

6Matrices And Determinants

If the determinant of a 3rd order matrix $ A $ is $ K $, then the sum of the determinants of the matrices $ A^4 $ and $ (A - A^4) $ is

7Matrices And Determinants

If $ \alpha, \beta, \gamma $ are the roots of $ \begin{bmatrix} 1 & -x & -2 \\ -2 & 4 & -x \\ -2 & 1 & -x \end{bmatrix} = 0 $, then $ \alpha \beta + \beta \gamma + \gamma \alpha = $

8Matrices And Determinants

If $A=\left[\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right], B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$ and $C=\left[\begin{array}{cc}0 & i \\ i & 0\end{array}\right]$, then

9Matrices And Determinants

Let $A=\left(\begin{array}{l}0 \\ -6 \\ 8\end{array}\right), B=\left(\begin{array}{ccc}3 & 5 & -7 \\ 0 & -1 & 8 \\ 6 & -1 & 0\end{array}\right)$ and $X=\left(\begin{array}{l}x \\ y \\ z\end{array}\right)$. If $D=[\alpha \beta \gamma]^T$ is ...

10Matrices And Determinants

If $S=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right]$ and $A=\frac{1}{2}\left[\begin{array}{lll}b+c & c-a & b-a \\ c-b & c+a & a-b \\ b-c & a-c & a+b\end{array}\right]$, then $S A S^{-1}=$

11Matrices And Determinants

If $A=\left\{\left(\begin{array}{ll}a & b \\ c & d\end{array}\right): a, b, c, d \in\{-1,1\}\right\}$, then the number of singular matrices in $A$ is

12Matrices And Determinants

If $\operatorname{det}(A B)=(\operatorname{det} A)(\operatorname{det} B)$ and $A$ is a non-singular matrix of order $3 \times 3$, then $\operatorname{det}(\operatorname{adj} A)=$

13Matrices And Determinants

Let $B=\left(\begin{array}{ccc}2 & 6 & 4 \\ 1 & 0 & 1 \\ -1 & 1 & -1\end{array}\right)$ and $C=\left(\begin{array}{ccc}-1 & 0 & 1 \\ 1 & 1 & 3 \\ 2 & 0 & 2\end{array}\right)$. If a matrix $A$ is such that $B A C=I$, then $A^{-1}=$

14Matrices And Determinants

If the co-factors of the elements 3, 7 and 6 of the matrix $\left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6\end{array}\right]$ are $a, b$ and $c$ respectively, then $\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\be...

15Matrices And Determinants

Let $$G(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$$. If \(x+y=0\) then \(G(x) G(y)=\)

16Matrices And Determinants

If $$A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$$, then \(\operatorname{det}\left(A^6+B^6\right)=\)

17Matrices And Determinants

If \(A X=D\) represents the system of simultaneous linear equations \(x+y+z=6, 5 x-y+2 z=3\) and \(2 x+y-z=-5\), then (Adj \(A\)) \(D=\)

18Matrices And Determinants

If $$A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]$$, then \(A A^T\) is a

19Matrices And Determinants

If \(\frac{x^4+24 x^2+28}{\left(x^2+1\right)^3}=\frac{A x+B}{x^2+1}\) \(+\frac{C x+D}{\left(x^2+1\right)^2}+\frac{E x+F}{\left(x^2+1\right)^3},\) then the value of \(A+B+C+D+E+F=\)

20Matrices And Determinants

If $$A=\frac{1}{7}\left[\begin{array}{ccc}3 & -2 & 6 \\ -6 & -3 & 2 \\ -2 & 6 & 3\end{array}\right]$$, then

21Matrices And Determinants

Let $$A=\left[\begin{array}{ccc}5 & \sin ^2 \theta & \cos ^2 \theta \\ -\sin ^2 \theta & -5 & 1 \\ \cos ^2 \theta & 1 & 5\end{array}\right]$$. Then, maximum value of \(\operatorname{det}(A)\) is

22Matrices And Determinants

If $$A=\left[\begin{array}{cc}\alpha^2 & 5 \\ 5 & -\alpha\end{array}\right]$$ and \(\operatorname{det}\left(A^{10}\right)=1024\), then \(\alpha=\)

23Matrices And Determinants

For \(i=1,2,3\) and \(j=1,23\)
If \(a_i^2+b_i^2+c_i^2=1, a_i a_j+b_i b_j+c_i c_j=0, \forall i \neq j\)
and $$A=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$$, then $$\operatorname{det}\left...

24Matrices And Determinants

If $$A=\left[\begin{array}{ccc}1 & 0 & 0 \\ a & -1 & 0 \\ b & c & 1\end{array}\right]$$ is such that \(A^2=I\), then

25Matrices And Determinants

If \(a, b, c\) are respectively the 5 th, 8 th, 13 th terms of an arithmetic progression, then $$\left|\begin{array}{ccc}a & 5 & 1 \\ b & 8 & 1 \\ c & 13 & 1\end{array}\right|=$$

26Matrices And Determinants

Let $$A=\left[\begin{array}{ccc}-2 & x & 1 \\ x & 1 & 1 \\ 2 & 3 & -1\end{array}\right]$$. If the roots of the equation \(\operatorname{det} A=0\) are \(l, m\) then \(l^3-m^3=\)

27Matrices And Determinants

If $$A=\left[\begin{array}{cc}2 & -3 \\ -4 & 1\end{array}\right]$$, then \(\left(A^T\right)^2+(12 A)^T=\)

28Matrices And Determinants

What is the value of $$\left|\begin{array}{ccc}a & b & c \\ a-b & b-c & c-a \\ b+c & c+a & a+b\end{array}\right|$$ ?

29Matrices And Determinants

If $$\left[\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right]\left[\begin{array}{cc}1 & \tan \theta \\ -\tan \theta & 1\end{array}\right]^{-1} =\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right]$$, then

30Matrices And Determinants

If \(A, B\) and \(C\) are the angles of a triangle, then the system of equations
\(-x+y \cos C+z \cos B=0, x \cos C-y+z \cos A=0\) and \(x \cos B+y \cos A-z=0\)

31Matrices And Determinants

The trace of the matrix $$A=\left[\begin{array}{ccc}1 & -5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{array}\right]$$ is

32Matrices And Determinants

If $$f(x)=\left|\begin{array}{ccc}x & x^2 & x^3 \\ 1 & 2 x & 3 x^2 \\ 0 & 2 & 6 x\end{array}\right|$$, then the ratio \(f^{\prime \prime}(x): f^{\prime}(x)\) is equal to

33Matrices And Determinants

Let A be a \(n\times n\) matrix such that A is upper-triangular. Then, \(adj (A)\) is equal to

34Matrices And Determinants

If $$\left|\begin{array}{lll}x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3\end{array}\right|=0$$ and \(x, y\) and \(z\) are all distinct, then \(x y z\) is equal to

35Matrices And Determinants

If $$A=\left[\begin{array}{llll}\sqrt{2020} & \sqrt{2021} & \sqrt{2021} & \sqrt{2023} \\ \sqrt{4040} & \sqrt{4042} & \sqrt{4044} & \sqrt{4046} \\ \sqrt{6060} & \sqrt{6063} & \sqrt{6066} & \sqrt{6069} \\ \sqrt{8080} & \sqrt{8084} & \sqrt{808...

36Matrices And Determinants

If \(k \in R\) and $$\operatorname{det} A=\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|=k$$, then $$\operatorname{det} B=\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2+2 a_1 & b_2+2 b_1 & c_...

37Matrices And Determinants

If \(\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\mathbf{c}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}\), then the value of $$\left|\...

38Matrices And Determinants

If \(a_1, a_2, \ldots . a_9\) are in GP, then $$\left|\begin{array}{lll}\log a_1 & \log a_2 & \log a_3 \\ \log a_4 & \log a_5 & \log a_6 \\ \log a_7 & \log a_8 & \log a_9\end{array}\right|$$ is equal to

39Matrices And Determinants

The rank of the matrix $$\left[\begin{array}{ccc}4 & 2 & (1-x) \\ 5 & k & 1 \\ 6 & 3 & (1+x)\end{array}\right]$$ is 1 , then,

40Matrices And Determinants

If $$A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right], 10 B=\left[\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3\end{array}\right]$$ and \(B=A^{-1}\), then the value of \(\alpha\) is

41Matrices And Determinants

Let a and b be non-zero real numbers such that \(ab=5/2\) and given \(A = \left[ {\matrix{ a & { - b} \cr b & a \cr } } \right]\) and \(A{A^T} = 20I\) (\(l\) is unit matrix), then the equation whose roots are a and b is

42Matrices And Determinants

The equation whose roots are the values of the equation \(\left| {\matrix{ 1 & { - 3} & 1 \cr 1 & 6 & 4 \cr 1 & {3x} & {{x^2}} \cr } } \right| = 0\) is

43Matrices And Determinants

Let \(a, b\) and \(c\) be such that \(b+c \neq 0\) and
$$\begin{aligned}
& \left|\begin{array}{ccc}
a & a+1 & a-1 \\
-b & b+1 & b-1 \\
c & c-1 & c+1
\end{array}\right| \\
& +\left|\begin{array}{ccc}
a+1 & b+1 & c-1 \\
a-1 & b-1 & c+1 \\
(-1...

44Matrices And Determinants

$$A=\left[\begin{array}{ccc}a^2 & 15 & 31 \\ 12 & b^2 & 41 \\ 35 & 61 & c^2\end{array}\right]$$ and $$B=\left[\begin{array}{ccc}2 a & 3 & 5 \\ 2 & 2 b & 8 \\ 1 & 4 & 2 c-3\end{array}\right]$$ are two matrices such that the sum of the princi...

45Matrices And Determinants

Let \(A, B, C, D\) be square real matrices such that \(C^T=D A B, D^{\mathrm{T}}=A B C\) and \(S=A B C D\), then \(S^2\) is equal to

46Matrices And Determinants

The value of $$\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|$$ is