Matrices & Determinants - Algebra PYQs | Mathematics

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

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

\(A=\left[\begin{array}{ccc}a^2 & 15 & 31 \\ 12 & b^2 & 41 \\ 35 & 61 & c^2\end{array}\right]\) 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 principal diagonal elements of both \(A\) and \(B\) are equal, then the product of the principal diagonal elements of \(B\) is

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)^{n+2} a & (-1)^{n-1} b & (-1)^n c \[\end{array}\right|=0 \text {, } \\\] & \end{aligned}\)

then the value of \(n\) is

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

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

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

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,

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

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 \(\left|\begin{array}{ccc}\mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{b} \cdot \mathbf{a} & \mathbf{b} \cdot \mathbf{b} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{c} \cdot \mathbf{a} & \mathbf{c} \cdot \mathbf{b} & \mathbf{c} \cdot \mathbf{c}\end{array}\right|\) is equal to

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 \(\operatorname{det} B=\left|\begin{array}{ccc}a_1 & b_1 & c_1 \\ a_2+2 a_1 & b_2+2 b_1 & c_2+2 c_1 \\ a_3 & b_3 & c_3\end{array}\right|\) is equal to

If \(A=\left[\begin{array}{llll}\sqrt{2020} & \sqrt{2021} & \sqrt{2021} & \sqrt{2023} \\ \sqrt{4040} & \sqrt{4042} & \sqrt{4044} & \sqrt{4046} \\ \sqrt{6060} & \sqrt{6063} & \sqrt{6066} & \sqrt{6069} \\ \sqrt{8080} & \sqrt{8084} & \sqrt{8088} & \sqrt{8092}\end{array}\right]\), then the rank of \(A\) is

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

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

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

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 \(x \cos B+y \cos A-z=0\)

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

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

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

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