Matrices & Determinants Questions - JEE

If $B B' = A^{-1} A' (A^{-1} A')' $, and simplifying, which of the following equals $BB'$?

Let $A = \begin{pmatrix}1&2&2\\2&1&1\\2&2&1\end{pmatrix}$. If $A$ is a zero divisor of $x^2 - 4x - 5$, find $\text{Tr}(A^3)$.

If $A = \begin{bmatrix} 2 & 0 & 0 \\ 2 & 2 & 0 \\ 2 & 2 & 2 \end{bmatrix}$, then $adj(adj A)$ is equal to -

If $A$ and $B$ are matrices of order $m \times n$ and $n \times m$ respectively, then order of matrix $B^T(A^T)^T$ is -

Let $A = \begin{pmatrix} \cos(\pi/5) & -\sin(\pi/5) \\ \sin(\pi/5) & \cos(\pi/5) \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. If $\displaystyle\sum_{r=1}^{4}\bigl(B^2 - B\cdot A^{9r} - A^r\cdot B + A^{10r}\bigr) = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then $a^{10} + b^{10} + c^{10} + d^{10} =$

Let $A$ and $B$ be $3\times 3$ invertible matrices such that $A^2B=BA$ and $P=A^{-1}$. Then the value of $P^3B^3A^3$ is:

The least value of the product xyz for which the determinant $\begin{vmatrix} x & 1 & 1 \\ 1 & y & 1 \\ 1 & 1 & z \end{vmatrix}$ is non-negative, is

Given matrix $A = \begin{bmatrix} 1 & x & 1 \\ x & 2 & y \\ 1 & y & 3 \end{bmatrix}; B = \begin{bmatrix} 3 & -3 & z \\ -3 & 2 & -3 \\ z & -3 & 1 \end{bmatrix}$. Obtain $x, y, z$ if the matrix $AB$ is symmetric.

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

If $\det(A)$=k, then det(A\)^3A⁻^3) equals:

Let $T=\{\alpha_1,\alpha_2,\beta\}$ be the 3 distinct roots of $x^2+x-1=0$ (note: a quadratic has at most 2 roots; here the set $T$ includes specific values). For a $3\times 3$ matrix $M=(a_{ij})$, let $R_i=a_{i1}+a_{i2}+a_{i3}$ and $C_j=a_{1j}+a_{2j}+a_{3j}$. Match entries in List-I with List-II: P) Number of $M$ with all entries in $T$ such that $R_i=C_j=0$ for all $i,j$ Q) Number of symmetric $M$ with all entries in $T$ such that $C_j=0$ for all $j$ R) Skew-symmetric $M$ with $a_{ij}\in T$ for $i>j$ — number of solutions to $M(x,y,z)^T=(-a,a,0)^T$ S) $M$ with all entries in $T$, $R_i=0$ for all $i$ — absolute value of $\det(M)$ List-II: 1)1, 2)12, 3)∞, 4)6, 5)0

**Paragraph:** Let $\alpha,\beta,\gamma$ be real roots of $x^3+ax^2+bx+c=0$ ($a,b,c\in\mathbb{R}$, $a,b\neq 0$). If the system $\alpha u+\beta v+\gamma w=0$, $\beta u+\gamma v+\alpha w=0$, $\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the minimum value of $x(3x+2a)+b+1$ is: A) $5$\quad B) $3$\quad C) $1$\quad D) $0$

If the system of linear equations$x + ay + z = 3$$x + 2y + 2z = 6$$x + 5y + 3z = b$has no solution, then

Let $A = \begin{pmatrix} \cos(\pi/5) & -\sin(\pi/5) \\ \sin(\pi/5) & \cos(\pi/5) \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. If $\displaystyle\sum_{r=1}^{4}\bigl(B^2 - B\cdot A^{9r} - A^r\cdot B + A^{10r}\bigr) = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then $a^{10} + b^{10} + c^{10} + d^{10} =$

Let $T=\{\alpha_1,\alpha_2,\beta\}$ be the 3 distinct roots of $x^2+x-1=0$ (note: a quadratic has at most 2 roots; here the set $T$ includes specific values). For a $3\times 3$ matrix $M=(a_{ij})$, let $R_i=a_{i1}+a_{i2}+a_{i3}$ and $C_j=a_{1j}+a_{2j}+a_{3j}$. Match entries in List-I with List-II: P) Number of $M$ with all entries in $T$ such that $R_i=C_j=0$ for all $i,j$ Q) Number of symmetric $M$ with all entries in $T$ such that $C_j=0$ for all $j$ R) Skew-symmetric $M$ with $a_{ij}\in T$ for $i>j$ — number of solutions to $M(x,y,z)^T=(-a,a,0)^T$ S) $M$ with all entries in $T$, $R_i=0$ for all $i$ — absolute value of $\det(M)$ List-II: 1)1, 2)12, 3)∞, 4)6, 5)0

**Paragraph:** Let $\alpha,\beta,\gamma$ be real roots of $x^3+ax^2+bx+c=0$ ($a,b,c\in\mathbb{R}$, $a,b\neq 0$). If the system $\alpha u+\beta v+\gamma w=0$, $\beta u+\gamma v+\alpha w=0$, $\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the minimum value of $x(3x+2a)+b+1$ is: A) $5$\quad B) $3$\quad C) $1$\quad D) $0$

**Paragraph (continued):** Consider the system $x+2y-3z=a$; $2x+6y-11z=b$; $x-2y+7z=c$. Then the system: A) has a unique solution when $5a=2b+c$ B) has infinite number of solutions when $5a=2b+c$ C) has no solution for all $a,b,c$ D) has a unique solution for all $a,b,c$

If $P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$, $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $Q = PAP^T$ and $x = P^T Q^{2005} P$, then $x$ is equal to -

$\begin{bmatrix} 1 & -\tan \theta/2 \\ \tan \theta/2 & 1 \end{bmatrix} \begin{bmatrix} 1 & \tan \theta/2 \\ -\tan \theta/2 & 1 \end{bmatrix}^{-1}$ is equal to -

Let $\phi(x) = \begin{vmatrix} x+a & x+b & x+a-c \\ x+b & x+c & x+1 \\ x+c & x+d & x+b-d \end{vmatrix}$ and $\int_0^2 \phi(x)\,dx = -16$, where $a, b, c$ and $d$ are in AP, then the common difference of the AP is equal to

If $A = \begin{bmatrix} 1 & 3 \\ 3 & 2 \\ 2 & 5 \end{bmatrix}$ & $B = \begin{bmatrix} -1 & -2 \\ 0 & 5 \\ 3 & 1 \end{bmatrix}$ and $A + B - D = O$ (zero matrix), then $D$ matrix will be-

If $\det(A)$=k, then det(A\)\cdotI) equals:

The number of values of $\theta\in[0,\pi]$ for which the system $3x-y+3z=2$, $x+y+4z=1$, $-6x+y+\lambda z=-3$ has unique solution is

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

Let $A=\begin{pmatrix}1&0&0\\\sqrt{a}&1&0\\a\sqrt{a}&\sqrt{b}&1\end{pmatrix}$; $a,b\in\mathbb{R}^+$. If for some $n\in\mathbb{N}$, $A^n=\begin{pmatrix}1&0&0\\72&1&0\\3600&72&1\end{pmatrix}$, then the number of triangles formed by joining the vertices of an $n$-sided polygon having no side common with the polygon is

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 $\Delta = \begin{vmatrix} 2a & 3a + 2x^2 & 4a + 3x^2 + 2x^4 \\ 2a & 3a + 2x^2 & 4a + 3x^2 + 2x^4 \\ 3a & 6a + 3x^2 & 10a + 6x^2 + 3x^4 \end{vmatrix}$

Which of the following matrices is NOT left inverse of matrix $\begin{bmatrix} 1 & -1 \\ 1 & 1 \\ 2 & 3 \end{bmatrix}$?

Consider, $A = \begin{bmatrix} a & 2 & 1 \\ 0 & b & 0 \\ 0 & -3 & c \end{bmatrix}$, where $a$, $b$ and $c$ are the roots of the equation $x^3 - 3x^2 + 2x - 1 = 0$. If matrix $B$ is such that $AB = BA$, $A + B - 2I \neq O$ and $A^2 - B^2 = 4I - 4B$, then find the value of $\det(B)$.

If θ ∈ R, then maximum value of $\Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+\sin\theta & 1 \\ 1 & 1 & 1+\cos\theta \end{vmatrix}$ is

If the system of linear equations x - 4y + 7z = g 3y - 5z = h -2x + 5y - 9z = k is consistent, then :

If $A$ and $B$ are square matrices of size $n \times n$ such that $A^2 - B^2 = (A - B)(A + B)$, then which of the following will be always true?

Let $ A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix} $ and $ B = A^{20} $. Then the sum of the elements of the first column of $ B $ is

Let $P = \begin{pmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{pmatrix}$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$, then $\frac{q_{31} + q_{32}}{q_{21}}$ equals

Number of 3 × 3 symmetric matrices which can be formed by three '0', three '1' & three '-1' only, is

If A = $\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ & A$^n$ = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, (where n $\ge$ 2 & n $\in$ N), then -

For Problems 4–6If $A$ and $B$ are two square matrices of order $3 \times 3$ which satisfy $AB = A$ and $BA = B$, then$(A + I)^5$ is equal to (where $I$ is identity matrix)

Let A be a square matrix such that $A(\text{adj. }A) = \begin{bmatrix}4 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 4\end{bmatrix}$. Find the values of:(i) |adj. A|

If $A$ and $B$ are symmetric matrices of the same order and $X = AB + BA$ and $Y = AB - BA$, then $(XY)^T$ is equal to

The number of $3 \times 3$ matrices $A$ whose entries are either 0 or 1 and for which the system $A \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ has exactly two distinct solutions is

If $\alpha, \beta, \gamma$ are the angles of a triangle and the system of equations \[\cos(\alpha-\beta)x + \cos(\beta-\gamma)y + \cos(\gamma-\alpha)z = 0\] \[\cos(\alpha+\beta)x + \cos(\beta+\gamma)y + \cos(\gamma+\alpha)z = 0\] \[\sin(\alpha+\beta)x + \sin(\beta+\gamma)y + \sin(\gamma+\alpha)z = 0\] has non-trivial solutions, then triangle is necessarily

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all nonsingular matrices of the form $\begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \theta & 1 \end{bmatrix}$, where each of $a, b,$ and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

If $P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$ is the adjoint of a 3 × 3 matrix A and |A| = 4, then $\alpha$ is equal to:

First row of a matrix A is [1 3 2]. If adj. $A = \begin{bmatrix} -2 & 4 & \alpha \\ -1 & 2 & 1 \\ 3\alpha & -5 & -2 \end{bmatrix}$, then det.(A) is

Let A and B be two $2 \times 2$ matrices. Consider the statements:(i) $AB = O \Rightarrow A = O$ or $B = O$(ii) $AB = I_2 \Rightarrow A = B^{-1}$(iii) $(A+B)^2 = A^2 + 2AB + B^2$Then

If A and B are two non-singular matrices of order 3 such that $AA^T = 2I$ and $A^{-1} = A^T - A \cdot \text{adj}(2B^{-1})$, then $\det(B)$ is equal to

If $A_1, B_1, C_1, \ldots$ are, respectively, the cofactors of the elements $a_1, b_1, c_1, \ldots$ of the determinant $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$, $\Delta \ne 0$, then the value of $\begin{vmatrix} B_2 & C_2 \\ B_3 & C_3 \end{vmatrix}$ is equal to

If a2 + b2 + c2 = -2 and f(x) = <mfenced open="|

If a, b, c are in AP, then x+1x+2x+ax+2x+3x+bx+3x+4x+c equals -

Matrices & Determinants Questions (2045)