Determinants Questions (2072)

The system of linear equations\(x + \lambda y - z = 0\)\(\lambda x - y - z = 0\)\(x + y - \lambda z = 0\)has a non-trivial solution for

Find the number of A = [aij]2x2 satisfying aij is 1 or -1 and a11a21 + a12a22 = 0.

The number of values of x for which the matrix \(A = \begin{pmatrix} 3-x & 2 & 2 \\ 2 & 4-x & 1 \\ -2 & -4 & -1-x \end{pmatrix}\) is singular, is

If the system of equation 2x + y - z = 5 2x - 5y + λz = μ x + 2y - 5z = 7 has infinitely many solutions, then (λ + μ)2 + (λ - μ)2 is equal to

If \(\mathbf{A} = \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix}\), then A is a/an

If the system of equations x + y - 3 = 0, (1 + K)x + (2 + K)y - 8 = 0 & x - (1 + K)y + (2 + K) = 0 is consistent then the value of K may be -

If determinant is diagonal matrix, its value is:

\(a, b, c\) are distinct real numbers, not equal to one. If \(ax + y + z = 0\), \(x + by + z = 0\), and \(x + y + cz = 0\) have a non-trivial solution, then the value of \(\dfrac{1}{1-a} + \dfrac{1}{1-b} + \dfrac{1}{1-c}\) is equal to

If the trivial solution is the only solution of the system of equations\(x + ky + z = 0\)\(kx + 3y + kz = 0\)\(3x + y + z = 0\)Then, the set of values of \(k\) is

Let \(M\) denote the matrix \(\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}\), where \(i^2 = -1\), and let \(I\) denote the identity matrix \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\). Then the matrix \(I + M + M^2 + M^3 + M^4 + \ldots + M^{2010}\) is equal to:

Find the sum of all positive integral values of a for which every solution to the system of equation x + ay = 3 and ax + 4y = 6 satisfy the inequalities x > 1, y > 0.

Let f(x)=1x-1x2(x-1)(x-1)(x-2)x(x-1)3(x-1)(x-2)(x-1)(x-2)(x-3)x(x-1)(x-2) & Dr=1102r70173r+1111. The value of ∑r=110Dr is

Let a1, a2, a3, ..., a10 be in G.P. with ai > 0 for i = 1,2,..., 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which . Then the number of elements in S, is :

For each real number x such that -1 A(x) = ⎡⎣1 -x⎤⎦ and let y and z = x/(1-x) and 1/(1-x). Then

Let us consider the following matrix: \[\begin{bmatrix} 1 & X & X \\ X & 1 & X \\ X & X & 1 \end{bmatrix}\] which are six non-singular matrices because six blanks (i.e., \(X\)) shall be filled by five zeros and one 1. In the same manner, we have the matrix: \[\begin{bmatrix} X & X & 1 \\ X & 1 & X \\ 1 & X & X \end{bmatrix}\] which are six non-singular matrices. Therefore, in the required case, there are more than 7. How many such matrices are possible?

Let A = [aij] be a 3x3 matrix such that AT = A and det(A) = 0. If the sum of the diagonal elements of A is 6 and the sum of the squares of the diagonal elements is 14, then the possible value(s) of det(A + I) is/are

The number of values of x in the closed interval [−4, −1], for which the matrix \(\begin{pmatrix} 3 & -1+x & 2 \\ 3 & -1 & x+2 \\ x+3 & -1 & 2 \end{pmatrix}\) is singular, is

If \(f(x) = \begin{vmatrix} 3 & 3x & 3x^2 + 2a^2 \\ 3x & 3x^2 + 2a^2 & 3x^3 + 6a^2x \\ 3x^2 + 2a^3 & 3x^3 + 6a^2x & 3x^4 + 12a^2x^2 + 2a^4 \end{vmatrix}\), then which is true?

Let p and p + 2 be prime numbers and let Δ = <mfenced open="|

Let $A = \begin{pmatrix}1&1\\1&1\end{pmatrix}$. If $\det(A^n - I) = 1 - \lambda^n$, find $\lambda$.

Question 83: Consider the determinant $\Delta = \begin{vmatrix} a_1 + b_1 x^2 & a_1 x^2 + b_1 & c_1 \\ a_2 + b_2 x^2 & a_2 x^2 + b_2 & c_2 \\ a_3 + b_3 x^2 & a_3 x^2 + b_3 & c_3 \end{vmatrix} = 0$, where $a_i, b_i, c_i \in \mathbb{R}$ and $x \in \mathbb{R}$.

The value of the determinant of a matrix is given by the answer key.

If det(\(A\)) = 0, then det(\(A^2\)) is:

Let $A = [a_{ij}]_{2 \times 2}$ where $a_{ij} \neq 0$ for all $i, j$ and $A^2 = I$. Let $a$ be the sum of all diagonal elements of $A$ and $b = |A|$, then $3a^2 + 4b^2$ is equal to

The value of the determinant of a matrix is given by the expression. If the matrix is singular, what is the value of the determinant?

If A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}, then A^2 - 4A - 5I is equal to

If <mfenced open="|

If \(\det(A)\)=k, then det((A\)^T)^2A^{-1}) equals:

If determinant \(\Delta\) = 0, then rank of matrix is:

If \(2s = a + b + c\) and the determinant \(\begin{vmatrix} a^2 & (s-a)^2 & (s-a)^2 \\ (s-b)^2 & b^2 & (s-b)^2 \\ (s-c)^2 & (s-c)^2 & c^2 \end{vmatrix} = ks^3(s-a)(s-b)(s-c)\), then the numerical quantity \(k\) should be

If A = \begin{pmatrix} 4 & 1 \\ 7 & 2 \end{pmatrix} and B = \begin{pmatrix} 2 & -1 \\ 7 & 4 \end{pmatrix}, then B^T A^T is

If \(\begin{bmatrix} 1/25 & 0 \\ x & 1/25 \end{bmatrix} = \left( \begin{bmatrix} 5 & 0 \\ -a & 5 \end{bmatrix}^{-1} \right)^2\), then the value of x is

If the system of linear equations 2x + 2ay + az = 0 2x + 3by + bz = 0 2x + 4cy + cz = 0 where a, b, c ∈ R are non-zero and distinct; has a non-zero solution, then :

The number of $3\times2$ matrices $A$, which can be formed using the elements of the set $\{-2,-1,0,1,2\}$ such that the sum of all the diagonal elements of $A^TA$ is 5, is

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices: $T_p = \left\{ A = \begin{bmatrix} a & b \\ c & a \end{bmatrix} : a, b, c \in \{0, 1, 2, \dots, p-1\} \right\}$. The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\det(A)$ divisible by $p$ is -

Let $M$ and $N$ be square matrices of the same order satisfying $MN = M$ and $NM = N$. Then $(M^{2024} + N^{2024})^{2025}$ is equal to

The number of values of \(k\) for which the system of equations\((k+1)x + 8y = 4k\)\(kx + (k+3)y = 3k - 1\)has no solution, is

Let A + 2B = ⎡⎣1 2 0⎤⎦ and 2A - B = ⎡⎣2 -1 5⎤⎦ and 6 -3 3 ⎤⎦ and 2 -1 6 ⎤⎦. If tr(A) denotes the sum of all diagonal elements of the matrix A, then tr(A) - tr(B) has value equal to(JEE Main 2021)

If the trivial solution is the only solution of the system of equations $x - ky + z = 0$, $kx + 3y - kz = 0$, $3x + y - z = 0$, then the set of all values of $k$ is :

If $x > m, y > n, z > r$ ($x, y, z > 0$) such that $\begin{vmatrix} x & n & r \\ m & y & r \\ m & n & z \end{vmatrix} = 0$, then the value of $\frac{x}{x-m} + \frac{y}{y-n} + \frac{z}{z-r}$ is

If $\begin{vmatrix} 3^2+k & 4^2 & 3^2+3+k \\ 4^2+k & 5^2 & 4^2+4+k \\ 5^2+k & 6^2 & 5^2+5+k \end{vmatrix} = 0$, then the value of $k$ is

Solve the system $x+y+z=6$, $x-y+z=2$, $2x+y-z=1$ using matrix method.

If $\begin{vmatrix} x+3 & 1 & -2 \\ 3 & -2 & 1 \\ -x & -3 & 3 \end{vmatrix} = 0$, find $x$.

If X, Y and Z are positive numbers such that Y and Z have respectively 1 and 0 at their unit's place and \begin{vmatrix} X & 4 & 1 \\ Y & 0 & 1 \\ Z & 1 & 0 \end{vmatrix} is divisible by 10, then X has at its unit's place:

The digits A, B, C are such that the three digit numbers A88, 6B8, 86C are divisible by 72. The determinant \begin{vmatrix} A & 6 & 8 \\ 8 & B & 6 \\ 8 & 8 & C \end{vmatrix} is divisible by

If in the determinant $\begin{vmatrix} x & 3 & 3 \\ 3 & 3 & x \\ 2 & 3 & 3 \end{vmatrix}$, $C_{11} = C_{22}$, where $C_{ij}$ is cofactor of element $a_{ij}$ then $x =$

Find the nature of solution for the given system of equations:$x + 2y + 3z = 1; 2x + 3y + 4z = 3; 3x + 4y + 5z = 0$

If \(p + q + r = 0 = a + b + c\), then the value of the determinant \(\begin{vmatrix} pa & qb & rc \\ qc & ra & pb \\ rb & pc & qa \end{vmatrix}\) is

Let \(A = \begin{bmatrix} 1 & 2 & -3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}\) and \(\text{adj } A = \begin{bmatrix} 1 & -2 & 7 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{bmatrix}\). Find the element \(A_{13}\) of \(A^{-1}\).

If \(a, b, c\) are non-zero real numbers and if the system of 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