Matrices & Determinants Questions - JEE

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A\,\text{adj}\,A = AA^T$, then $5a + b$ is equal to:

If $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$, then the matrix $A^{-50}$ when $\theta = \dfrac{\pi}{12}$, is equal to:

The system of linear equations$x + \mu y - z = 0$$\mu x - y - z = 0$$x + y - \mu z = 0$has a non-trivial solution for:

Let $\lambda$ be a real number for which the system of linear equations$x + y + z = 6$$4x + \lambda y - \lambda z = \lambda - 2$$3x + 2y - 4z = -5$has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation

If $x = a$, $y = b$, $z = c$ is a solution of the system of linear equations$x + 8y + 7z = 0$$9x + 2y + 3z = 0$$x + y + z = 0$such that the point $(a, b, c)$ lies on the plane $x + 2y + z = 6$, then $2a + b + c$ equals

Given $(A - 3I)(A - 5I) = 0$ for a matrix $A$, which gives $\alpha A + \beta A^{-1} = 4I$. Then $\alpha + \beta$ equals:

The sum of the elements of $U^{-1}$ is

Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $a, b, c, d \neq 0$. If $A^2 = A$, then the value of $|A|$ (determinant of $A$) is

If $A^2 - A + I = 0$ where $A$ is a square matrix and $I$ is the unit matrix of the same order then $A^{-1}$ is

If x, y and z are the integers in AP lying between 1 and 9 and x51, y41 and z31 are three digits numbers, the value of \begin{vmatrix} x & 51 \\ y & 41 \\ z & 31 \end{vmatrix} \div \begin{vmatrix} x & 5 \\ y & 4 \\ z & 3 \end{vmatrix} is

If $A \times \begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix} = 6I$ where $I$ is a unit matrix of order $2 \times 2$, then $A$ is

The system of linear equations$x + y + z = 2$$2x + 3y + 2z = 5$$2x + 3y + (a^2 - 1)z = a + 1$

Let $A = \begin{bmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{bmatrix}$. Which of the following is true?

Let $ A $ be a $ 3 \times 3 $ matrix such that $ A^2 - 5A + 7I = 0 $.Statement-I: $ A^{-1} = \dfrac{1}{7}(5I - A) $.Statement-II: The polynomial $ A^3 - 2A^2 - 3A + I $ can be reduced to $ 5(A - 4I) $.Then:

If $A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$, then which one of the following statements is not correct?

If $A$ is a $3 \times 3$ matrix such that $|5 \cdot \text{adj}\, A| = 5$, then $|A|$ is equal to

Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A - 3I)(A - 5I) = O$, where $I = I_3$ and $O = O_3$. If $\alpha A + \beta A^{-1} = 4I$, then $\alpha + \beta$ is equal to

If $A^k = 0$ (A is nilpotent with index k), $(I - A)^p = I + A + A^2 + \ldots + A^{k-1}$, thus p is,

If $\Delta$ = 7 and two rows are interchanged, new determinant is:

Given $A = \begin{bmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{bmatrix}$ where $\theta \in \left(\dfrac{3\pi}{4}, \dfrac{5\pi}{4}\right)$, then $\det(A)$ lies in the interval:

If determinant is non-zero, inverse exists because:

If $\phi(x) = \begin{vmatrix} x^2 + 5x + 3 & 2x + 5 & 3 \\ 3x^2 - x - 4 & 6x - 1 & 9 \\ 7x^2 + 6x + 9 & 14x + 6 & 21 \end{vmatrix} = ax^3 + bx^2 + cx + d$, then

If $a_1b_1c_1$, $a_2b_2c_2$, and $a_3b_3c_3$ are three-digit even natural numbers and $\Delta = \begin{vmatrix} c_1 & a_1 & b_1 \\ c_2 & a_2 & b_2 \\ c_3 & a_3 & b_3 \end{vmatrix}$, then $\Delta$ is

The number of $3 \times 3$ non-singular matrices, with four entries as 1 and all other entries as 0, is

If $f(x) = a + bx + cx^2$ and $\alpha, \beta, \gamma$ are the roots of the equation $x^2 = 1$, then $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ is equal to

Let $\vec{a}_r = x_r\hat{i} + y_r\hat{j} + z_r\hat{k}$, $r = 1, 2, 3$ be three mutually perpendicular unit vectors, then the value of $\begin{vmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \end{vmatrix}$ is equal to

If $A_r = \begin{vmatrix} 2^{r-1} & 2 \times 3^{r-1} & 4 \times 5^{r-1} \\ \alpha & \beta & \gamma \\ 2^n - 1 & 3^n - 1 & 5^n - 1 \end{vmatrix}$, then find the value of $\displaystyle\sum_{r=1}^{n} A_r$.

The value of the determinant \[\begin{vmatrix} (a_1-b_1)^2 & (a_1-b_2)^2 & (a_1-b_3)^2 & (a_1-b_4)^2 \\ (a_2-b_1)^2 & (a_2-b_2)^2 & (a_2-b_3)^2 & (a_2-b_4)^2 \\ (a_3-b_1)^2 & (a_3-b_2)^2 & (a_3-b_3)^2 & (a_3-b_4)^2 \\ (a_4-b_1)^2 & (a_4-b_2)^2 & (a_4-b_3)^2 & (a_4-b_4)^2 \end{vmatrix}\] is

If $a - b + c = 0$, then one root of $\begin{vmatrix} a-x & c & b \\ c & b-x & a \\ b & a & c-x \end{vmatrix} = 0$ is

For Problems 16–18Consider the polynomial function\[f(x) = \begin{vmatrix} (1+x)^a & (1+2x)^b & 1 \\ 1 & (1+x)^a & (1+2x)^b \\ (1+2x)^b & 1 & (1+x)^a \end{vmatrix}\]$a, b$ being positive integers.The constant term in $f(x)$ is

For Problems 22–24Consider the system of equations$x + y + z = 6$$x + 2y + 3z = 10$$x + 2y + \lambda z = \mu$The system has infinite solutions if

If $\begin{vmatrix} x^n & x^{n+2} & x^{2n} \\ 1 & x^a & a \\ x^{n+5} & x^{a+6} & x^{2n+5} \end{vmatrix} = 0, \forall x \in R$, where $n \in N$, then value of $a$ is

Prove that \[\begin{vmatrix} 1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1+d \end{vmatrix} = abcd\left(1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}\right).\] Hence, find the value of the determinant if $a, b, c, d$ are the roots of the equation $px^4 + qx^3 + rx^2 + sx + t = 0$.

The total number of distinct $x \in R$ for which \[\begin{vmatrix} x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{vmatrix} = 10\] is ___. (JEE Advanced 2016)

Which must be true: I) $A,B$ square matrices of same order, $O$ null, $I$ identity. If $AB=O$, then $(\det A)^2+(\det B)^2=0$. II) If $l_i,m_i,n_i$ $(i=1,2,3)$ are direction cosines of 3 mutual perpendicular vectors, then $A=\begin{pmatrix}l_1&m_1&n_1\\l_2&m_2&n_2\\l_3&m_3&n_3\end{pmatrix}$ is orthogonal. III) For two distinct lines with shortest distance $d\ne 0$, a point $A$ on line 1 and $B$ on line 2 with $AB=d$: $A$ and $B$ are unique.

If $A^2 - A + I = 0$, then the inverse of A is

A is a $2 \times 2$ matrix such that $A\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}-1\\2\end{bmatrix}$ and $A^2\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}$. The sum of the elements of A is

If $k \in R_0$, then $\det(\text{adj}(kI_n))$ is equal to

If a, b and c are non-zero real numbers and if the equations (a-1)x = y + z, (b-1)y = z + x, (c-1)z = x + y has a non-trivial solution, then ab + bc + ca equals

Let k be a positive real number and \[A = \begin{bmatrix} 2k-1 & 2\sqrt{k} & 2\sqrt{k} \\ 2\sqrt{k} & 1 & -2k \\ -2\sqrt{k} & 2k & -1 \end{bmatrix}\] and \[B = \begin{bmatrix} 0 & 2k-1 & \sqrt{k} \\ 1-2k & 0 & 2 \\ -\sqrt{k} & -2\sqrt{k} & 0 \end{bmatrix}.\] If $\det(\text{adj}\, A) + \det(\text{adj}\, B) = 10^6$, then $[k]$ is equal to ___. [Note: adj M denotes the adjoint of a square matrix M and $[k]$ denotes the largest integer less than or equal to $k$.]

The number of solutions of the matrix equation $X^2 = \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}$ is

The number of distinct real roots of the equation $\begin{vmatrix} \cos x & \sin x & \sin x \\ \sin x & \cos x & \sin x \\ \sin x & \sin x & \cos x \end{vmatrix} = 0$ in the interval $\left[-\dfrac{\pi}{4}, \dfrac{\pi}{4}\right]$ is

There are two possible values of A in the solution of the matrix equation $\begin{bmatrix} 2A+1 & -5 \\ -4 & A \end{bmatrix}^{-1} \begin{bmatrix} A-5 & B \\ 2A-2 & C \end{bmatrix} = \begin{bmatrix} 14 & D \\ E & F \end{bmatrix}$, where A, B, C, D, E and F are real numbers. The absolute value of the difference of the two solutions is

If $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} 1/2 & 1/2 & 1/2 \\ -4 & 3 & c \\ 5/2 & -3/2 & 1/2 \end{bmatrix}$, then the values of a and c are equal to

If $\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x + iy$, then

Matrices & Determinants Questions (2045)