(B) square matrix $A$ is invertible if and only if its determinant is non-zero,i.e.,$|A| \neq 0$.Given the matrix $A = \begin{bmatrix} \lambda & -1 &

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(B) square matrix $A$ is invertible if and only if its determinant is non-zero,i.e.,$|A| \neq 0$.Given the matrix $A = \begin{bmatrix} \lambda & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2 \end{bmatrix}$.Calculating the determinant $|A|$:$|A| = \lambda(0 \times 2 - 1 \times 1) - (-1)(-3 \times 2 - 1 \times (-1)) + 4(-3 \times 1 - 0 \times (-1))$$|A| = \lambda(0 - 1) + 1(-6 + 1) + 4(-3 - 0)$$|A| = -\lambda - 5 - 12$$|A| = -\lambda - 17$For the matrix to be invertible,$|A| \neq 0$:$-\lambda - 17 \neq 0$$-\lambda \neq 17$$\lambda \neq -17$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

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The matrix $\begin{bmatrix} \lambda & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2 \end{bmatrix}$ is invertible,if

A
$\lambda \neq -15$
B
$\lambda \neq -17$
C
$\lambda \neq -16$
D
$\lambda \neq -18$

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3 and 4 .Determinants and Matrices

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Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

If $-9$ is a root of the equation $\left| \begin{array}{ccc} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} \right| = 0$,then the other two roots are:

MediumIIT 1983

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$\Delta = \left| \begin{array}{ccc} a & a+b & a+b+c \\ 3a & 4a+3b & 5a+4b+3c \\ 6a & 9a+6b & 11a+9b+6c \end{array} \right|$ where $a = i, b = \omega, c = \omega^2$,then $\Delta$ is equal to

Medium

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The planes $x = cy + bz, y = az + cx, z = bx + ay$ pass through one line,if

Medium

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If $\alpha$ is a real root of the equation $x^3+6x^2+5x-42=0$,then the determinant of the matrix $\left[\begin{array}{ccc}\alpha-1 & \alpha+1 & \alpha+2 \\ \alpha-2 & \alpha+3 & \alpha-3 \\ \alpha+4 & \alpha-4 & \alpha+5\end{array}\right]$ is

MediumAP EAMCET 2025

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The area of a triangle with vertices $(K, 0), (4, 0), (0, 2)$ is $4$ square units. Find the value of $K$.

EasyKCET 2017

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The matrix $\begin{bmatrix} \lambda & -1 & 4 \\ -3 & 0 & 1 \\ -1 & 1 & 2 \end{bmatrix}$ is invertible,if

Similar Questions

If $-9$ is a root of the equation $\left| \begin{array}{ccc} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array} \right| = 0$,then the other two roots are:

$\Delta = \left| \begin{array}{ccc} a & a+b & a+b+c \\ 3a & 4a+3b & 5a+4b+3c \\ 6a & 9a+6b & 11a+9b+6c \end{array} \right|$ where $a = i, b = \omega, c = \omega^2$,then $\Delta$ is equal to

The planes $x = cy + bz, y = az + cx, z = bx + ay$ pass through one line,if

If $\alpha$ is a real root of the equation $x^3+6x^2+5x-42=0$,then the determinant of the matrix $\left[\begin{array}{ccc}\alpha-1 & \alpha+1 & \alpha+2 \\ \alpha-2 & \alpha+3 & \alpha-3 \\ \alpha+4 & \alpha-4 & \alpha+5\end{array}\right]$ is

The area of a triangle with vertices $(K, 0), (4, 0), (0, 2)$ is $4$ square units. Find the value of $K$.

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