Question:
Let $M$ be a $3 \times 3$ non-singular matrix with $det(M)=\alpha$. If $\left[M^{-1} adj(adj(M)]=K I\right.$, then the value of $K$ is
Let $M$ be a $3 \times 3$ non-singular matrix with $det(M)=\alpha$. If $\left[M^{-1} adj(adj(M)]=K I\right.$, then the value of $K$ is
Updated On: Jan 30, 2025
- 1
- $\alpha$
- $\alpha^2$
- $\alpha^3$
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Formulae:- $A ^{-1}=\frac{1}{| A |} adj( A )$
$A \cdot adj( A )=| A | I =adj( A ) \cdot A$
$adj(adj A)=| A |^{( n -2)} A$
Now, ${\left[ M ^{-1} adj(adj( M )]\right.}$
$=\frac{1}{| M |} adj( M ) \cdot| M |^{(3-2)} M$
$=adj( M ) \cdot M$
$=| M | I$
$=\alpha I$
$\therefore k =\alpha$
$A \cdot adj( A )=| A | I =adj( A ) \cdot A$
$adj(adj A)=| A |^{( n -2)} A$
Now, ${\left[ M ^{-1} adj(adj( M )]\right.}$
$=\frac{1}{| M |} adj( M ) \cdot| M |^{(3-2)} M$
$=adj( M ) \cdot M$
$=| M | I$
$=\alpha I$
$\therefore k =\alpha$
Was this answer helpful?
0
0
Top Questions on Properties of Determinants
- If \(\alpha \neq a\), \(\beta \neq b\), \(\gamma \neq c\) and \[ \begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0,\] then \[ \frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c} \] is equal to:
- JEE Main - 2024
- Mathematics
- Properties of Determinants
- If \[ A = \begin{bmatrix} -7 & 3 \\ 3 & -1 \end{bmatrix} \] then \( \det(A^5) \) is equal to:
- KEAM - 2024
- Mathematics
- Properties of Determinants
- Let \( A \) be a \( 3 \times 3 \) matrix with \( |A| = 7 \). If \( B = 3A \), then the value of \[ \left| \frac{{adj } A}{B} \right| \] is equal to
- KEAM - 2024
- Mathematics
- Properties of Determinants
Let A be an invertible matrix of size 4x4 with complex entries. If the determinant of adj(A) is 5,then the number of possible value of determinant of A is
- KEAM - 2023
- Mathematics
- Properties of Determinants
- If for some $c < 0$, the quadratic equation, $2cx^{2}-2\left(2c-1\right)x+3c^{2}=0$ has two distinct real roots, $\frac{1}{a}$ and $\frac{1}{b},$ then the value of the determinant. $\begin{vmatrix}1+a&1&1\\ 1&1+b&1\\ 1&1&1+c\end{vmatrix}$ is :
- JEE Main - 2020
- Mathematics
- Properties of Determinants
View More Questions
Questions Asked in BITSAT exam
- Inside a solenoid of radius \( 0.5 \) m, the magnetic field is changing at a rate of \( 50 \times 10^{-6} \) T/s. The acceleration of an electron placed at a distance of \( 0.3 \) m from the axis of the solenoid will be:
- BITSAT - 2024
- thermal properties of matter
In the given cycle ABCDA, the heat required for an ideal monoatomic gas will be:
- BITSAT - 2024
- specific heat capacity
- If \( a, c, b \) are in GP, then the area of the triangle formed by the lines \( ax + by + c = 0 \) with the coordinate axes is equal to:
- BITSAT - 2024
- Vectors
- A spherical ball of mass \( 20 \) kg is stationary at the top of a hill of height \( 100 \) m. It rolls down a smooth surface to the ground, then climbs up another hill of height \( 30 \) m and finally rolls down to a horizontal base at a height of \( 20 \) m above the ground. The velocity attained by the ball is:
- BITSAT - 2024
- Reflection Of Light By Spherical Mirrors
Evaluate the following limit: $ \lim_{n \to \infty} \prod_{r=3}^n \frac{r^3 - 8}{r^3 + 8} $.
- BITSAT - 2024
- limits and derivatives
View More Questions
Concepts Used:
Properties of Determinants
Properties of Determinants:
- Reflection Property: The value of the determinant remains unchanged if its rows and columns are interchanged.
- Switching Property: The interchanging of any two rows (or columns) of the Determinant changes its signs.
- All- Zero Property: The Determinants will be equivalent to zero if each term of rows and columns is zero.
- Proportionality (Repetition) Property: If all elements of a row (or column) are proportional or identical to the elements of some other row (or column), then the determinant is zero.
- Scalar Multiple Property: If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.
- Sum Property: If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.
- Property of Invariance: If to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation Ri → Ri + kRj or Ci → Ci + kCj
- Factor Property: If a Determinant Δ becomes 0 while considering the value of x = α, then (x - α) is considered as a factor of Δ
- Triangle Property: If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements.
Read More: Properties of Determinants