Determinants Questions (2072)

We have \[A = \begin{bmatrix}\cos^2\theta & \cos\theta\sin\theta\\ \cos\theta\sin\theta & \sin^2\theta\end{bmatrix}\] and \[B = \begin{bmatrix}\cos^2\phi & \cos\phi\sin\phi\\ \cos\phi\sin\phi & \sin^2\phi\end{bmatrix}.\] If \(AB = O\), then \(\theta - \phi\) is an odd multiple of how many degrees?

If sin2xcos2x4sin2x2tan2x2cos2x-sin2x-2cos4xtan2x2sin4x = a_0 + a_1(cos x) + a_2(cos^2 x) + ........ + a_n(cos^n x), then a_0 is -

If \(f(x) = \begin{vmatrix} 1+a^2x & (a+b^2)x & (1+c^2)x \\ (1+a^2)x & 1+b^2x & (1+c^2)x \\ (1+a^2)x & (1+b^2)x & 1+c^2x \end{vmatrix}\) and \(a^2+b^2+c^2 = -2\), then \(f(x)\) is a polynomial of degree:

The product of matrices \(A = \begin{bmatrix}\cos^2\theta & \cos\theta\sin\theta\\ \cos\theta\sin\theta & \sin^2\theta\end{bmatrix}\) and \(B = \begin{bmatrix}\cos^2\phi & \cos\phi\sin\phi\\ \cos\phi\sin\phi & \sin^2\phi\end{bmatrix}\) is a null matrix if \(\theta - \phi =\)

258. If \(A = \begin{bmatrix} a & x & y \\ x & b & z \\ y & z & c \end{bmatrix}\) where \(a, b, c, x, y, z \in \{1, 2, 3, 4, 5, 6\}\) and also \(a, b, c, x, y, z\) are distinct, then number of matrices in \(A\) with trace equal to 10 are:

Let A be a 2 × 2 matrix.Statement 1: adj(adj A) = AStatement 2: |adj A| = |A|

Let A be a 2 × 3 matrix whereas B be a 3 × 2 matrix. If det(AB) = 4, then find the value of det(BA).

Consider the matrix $f(x)=\begin{bmatrix}\cos x&-\sin x&0\\\sin x&\cos x&0\\0&0&1\end{bmatrix}$. Given below are two statements: Statement I: $f(-x)$ is the inverse of the matrix $f(x)$. Statement II: $f(x)f(y)=f(x+y)$. In the light of the above statements, choose the correct answer from the options given below

If \(w\) is a complex cube root of unity, then value of \(\Delta = \begin{vmatrix} a_1 + b_1 w & a_1 w^2 + b_1 & c_1 + b_1 \bar{w} \\ a_2 + b_2 w & a_2 w^2 + b_2 & c_2 + b_2 \bar{w} \\ a_3 + b_3 w & a_3 w^2 + b_3 & c_3 + b_3 \bar{w} \end{vmatrix}\) is

If \(A = \begin{bmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}\) be such that \(AB = \begin{bmatrix} 6 \\ 8 \end{bmatrix}\), then

Consider the system of equations\[2x + \lambda y + 6z = 8\] \[x + 2y + mz = 5\] \[x + y + 3z = 4\]The system of equations has infinitely many solutions if:

Let \[\begin{vmatrix} x & 2 & x \\ x^2 & x & 6 \\ x & x & 6 \end{vmatrix} = Ax^4 + Bx^3 + Cx^2 + Dx + E\]. Then the value of \(5A + 4B + 3C + 2D + E\) is equal to

If a, b, c are distinct and\[\begin{vmatrix} a & a^2 & a^4-1 \\ b & b^2 & b^4-1 \\ c & c^2 & c^4-1 \end{vmatrix} = 0\]then find the value of \(abc(a+b+c)\).

Given \[\begin{vmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{vmatrix}\] Find the value of \(x\) such that the determinant equals \((x+a+b+c)(a+b+c)^2\).

Given \(A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^2 & c^2 \end{bmatrix}\) and \(\det(A) \in [2, 16]\). If \(2, b, c\) are in A.P., then \(c\) lies in the interval:

The system of homogeneous equations \(\lambda x + (\lambda + 1)y + (\lambda - 1)z = 0\), \((\lambda + 1)x + \lambda y + (\lambda + 2)z = 0\), \((\lambda - 1)x + (\lambda + 2)y + \lambda z = 0\) has non-trivial solution for:

Find the maximum value of the determinant of an arbitrary 3 × 3 matrix A, each of whose entries \(a_{ij} \in \{-1, 1\}\).

Let $R=\begin{pmatrix}x&0&0\\0&y&0\\0&0&z\end{pmatrix}$ be a non-zero $3\times3$ matrix, where $x\sin\theta=y\sin\!\left(\theta+\dfrac{2\pi}{3}\right)=z\sin\!\left(\theta+\dfrac{4\pi}{3}\right)\neq0$, $\theta\in(0,2\pi)$. For a square matrix $M$, let trace$(M)$ denote the sum of all diagonal entries of $M$. Then, among the statements: (I) Trace$(R)=0$ (II) If trace$(\text{adj}(\text{adj}(R)))=0$, then $R$ has exactly one non-zero entry.

Find the value of x for which the matrix\[A = \begin{bmatrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}\] is inverse of \[B = \begin{bmatrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \end{bmatrix}\]

Consider the system of equations\[2x + \lambda y + 6z = 8\] \[x + 2y + mz = 5\] \[x + y + 3z = 4\]The system of equations has no solution if:

If \[\begin{vmatrix} a^2+\lambda^2 & ab+c\lambda & ca-b\lambda \\ ab-c\lambda & b^2+\lambda^2 & bc+a\lambda \\ ca+b\lambda & bc-a\lambda & c^2+\lambda^2 \end{vmatrix} \begin{vmatrix} \lambda & c & -b \\ -c & \lambda & a \\ b & -a & \lambda \end{vmatrix} = (1-a^2+b^2+c^2)^3\), then the value of \(\lambda\) is

If A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & -1 \end{pmatrix} is a matrix satisfying the equation AA^T = 9I, where I is 3×3 identity matrix, then (a, b) is equal to

If $A = \begin{bmatrix} 1 & -3 \\ 2 & -1 \end{bmatrix}$ and $A^2 - 4A + 10I = A$, then $k$ is equal to

If \(\phi(x) = \begin{vmatrix} 4x + 4 & (x+2)^2 & x^3 \\ 8x + 4 & 2(x+2)^2 & (x+1)^3 \\ 12x + 4 & 3(x+2)^2 & (x+13) \end{vmatrix}\), then which statement about \(\phi(x)\) is correct?

If \(a\), \(b\), \(c\) are positive and are the \(p\)th, \(q\)th, and \(r\)th terms, respectively, of a G.P., then \(\Delta = \begin{vmatrix} \log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1 \end{vmatrix}\) is

Let A = \(\begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}\) be a square matrix and C1, C2, C3 be three column matrices satisfying \(AC_1 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\), \(AC_2 = \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix}\), \(AC_3 = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}\), and B = [C1 C2 C3] is a matrix formed by these column matrices. The value of det(B-1) is

Let $A = \begin{bmatrix} 1 & -3 & 2 \\ 2 & 1 & -3 \\ 4 & -3 & -1 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 4 & 1 & 0 \\ 2 & 1 & 1 & 1 \\ 1 & -2 & 1 & 2 \end{bmatrix}$ & $C = \begin{bmatrix} 1 & 1 & -1 & -2 \\ 3 & -2 & -1 & -1 \\ 2 & -5 & -1 & 0 \end{bmatrix}$ be the matrices then, prove that in matrix multiplication cancellation law does not hold.

For $x > 0$, let $A = \begin{bmatrix} x+1 & 0 & 0 \\ 0 & \frac{1}{x} & 0 \\ 0 & 0 & 12 \end{bmatrix}$, $B = \begin{bmatrix} 0 & a_2 & a_3 \\ 0 & \frac{x}{3} & 0 \\ 0 & 0 & \frac{3}{x} \end{bmatrix}$ be two matrices and $C = AB + (AB)^2 + \cdots + (AB)^n$. Then $\operatorname{Tr}\left(\lim_{n \to \infty} C\right)$ is equal to

$A = \begin{bmatrix} 2 & -1 \\ -7 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 1 \\ 7 & 2 \end{bmatrix}$ then $B^T A^T$ is

Given \(x + 3y + 7z = 0\), \(-x + 4y + 7z = 0\) and \((\sin 3\theta)x + (\cos 2\theta)y + 2z = 0\). For non-trivial solutions, find the number of values of \(\theta\) in \([0, 2\pi]\).

If matrix A is orthogonal with rows R_1, R_2, R_3, and R_1 \times R_3 = 0 gives x + 4 + 2y = 0, and R_2 \times R_3 = 0 gives 2x + 2 - 2y = 0, find xy.

If \(\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c)\), where \(a, b\) and \(c\) are all different, then the determinant \(\begin{vmatrix} \frac{1}{(x-a)^2} & \frac{1}{(x-b)^2} & \frac{1}{(x-c)^2} \\ \frac{1}{(x-b)(x-c)} & \frac{1}{(x-c)(x-a)} & \frac{1}{(x-a)(x-b)} \end{vmatrix}\) vanishes when

If \(F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}\) and \(G(y) = \begin{bmatrix} \cos y & 0 & \sin y \\ 0 & 1 & 0 \\ -\sin y & 0 & \cos y \end{bmatrix}\), then \([F(x)\, G(y)]^{-1}\) is equal to

If \(A = \begin{bmatrix} a & b \\ b & a \end{bmatrix}\) and \(A^2 = \begin{bmatrix} \alpha & \beta \\ \beta & \alpha \end{bmatrix}\) then

Evaluate $\sum_{m=1}^{n} \Delta_m$ where $\Delta_m = \begin{vmatrix} 2r-1 & ^nC_r & 1 \\ m^2-1 & 2^m & m+1 \\ \sin^2(m^2) & \sin^2(m) & \sin^2(m+1) \end{vmatrix}$.

If $\begin{bmatrix} 5 & 1 & 4 \\ 7 & 0 & 2 \\ 1 & 3 & 5 \end{bmatrix} \begin{bmatrix} 3 & 6 & -7 \\ -6 & 2 & 4 \\ 1 & 6 & 3 \end{bmatrix} \begin{bmatrix} 9 & 7 & 1 \\ 1 & 6 & 3 \end{bmatrix} = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}$, then the value of $2|a_2 - b_1| + 3|a_3 - c_1| + 4|b_3 - c_2|$ is equal to

If \(\Delta = \begin{vmatrix} \sin x & \sin(x+h) & \sin(x+2h) \\ \sin(x+2h) & \sin x & \sin(x+h) \\ \sin(x+h) & \sin(x+2h) & \sin x \end{vmatrix}\), find \(\lim_{h \to 0} \frac{\Delta}{h^2}\).

Let $\begin{vmatrix}a&\sqrt{5}&\sqrt{7}\\\sqrt{3}&b&\sqrt{7}\\\sqrt{3}&\sqrt{5}&c\end{vmatrix}=0$, ($a\neq\sqrt{3}, b\neq\sqrt{5}, c\neq\sqrt{7}$) and $\dfrac{a}{a-\sqrt{3}}+\dfrac{b}{b-\sqrt{5}}+\dfrac{c}{c-\sqrt{7}}=\lambda$. If $a=2\sqrt{3}$, then the point $(b^2,c^2)$ may lie on the line

If f(x) = \begin{vmatrix} \sin x & \cos x & \sin x \\ \cos x & -\sin x & \cos x \\ x & 1 & 1 \end{vmatrix}, find the value of 2 \cdot \frac{f'(0)}{[f'(1)]^2}

Let \[A = \begin{bmatrix} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{bmatrix}\] Then which of the following is true?

For two \( 3 \times 3 \) matrices \( A \) and \( B \), let \( A + B = 2B' \) and \( 3A + 2B = I_3 \), where \( B' \) is the transpose of \( B \) and \( I_3 \) is \( 3 \times 3 \) identity matrix. Then

Let \(A\) be a matrix such that \(A\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}\) is a scalar matrix and \(|3A| = 108\). Then \(A^2\) equals

Let \(xC_i\), \(x^2C_i\) and \(x^3C_i\) (\(i \in \{1, 2, 3\}\)) be Binomial coefficients, where \(x \in \mathbb{N}\)and \[f(x) = \begin{vmatrix} xC_1 & xC_2 & xC_3 \\ x^2C_1 & x^2C_2 & x^2C_3 \\ x^3C_1 & x^3C_2 & x^3C_3 \end{vmatrix}\]Then \(f(x)\) is a polynomial of degree:

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The system of equations $x_1 - x_2 + x_3 = 2, 3x_1 - x_2 + 2x_3 = -6$ and $3x_1 + x_2 + x_3 = -18$ has

If \(\alpha\), \(\beta\), \(\gamma\) are constants, then the value of\[\sum_{r=1}^{n} S_r = \begin{vmatrix} \sum_{r=1}^{n} 2^{r-1} & \alpha & 2^{n-1} \\ \sum_{r=1}^{n} 2 \cdot 3^{r-1} & \beta & 3^{n-1} \\ \sum_{r=1}^{n} 4 \cdot 5^{r-1} & \gamma & 5^{n-1} \end{vmatrix}\]equals

If \(A_i = \begin{pmatrix} 2 - i & 3 - i \\ 3 & 2 \end{pmatrix}\), then \(\sum_{i=1}^{\infty} \det(A_i)\) is equal to

If the value of the determinant \(\begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix}\) is positive, then \((a, b, c > 0)\)

If A = $$\begin{bmatrix} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{bmatrix}$$, then A is

If \(\Delta\) = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and each element is multiplied by k, determinant becomes: