Question:
Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M^{-1}=\text{adj}(\text{adj} M)$, then which of the following statements is/are ALWAYS TRUE?
Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M^{-1}=\text{adj}(\text{adj} M)$, then which of the following statements is/are ALWAYS TRUE?
Updated On: Apr 28, 2025
- $M=I$
- $\text{det} M=1$
- $M^{2}=I$
- $(\text{adj} M)^{2}=I$
Show Solution
Verified By Collegedunia
The Correct Option is B, C, D
Solution and Explanation
(B) $\text{det} M=1$
(C) $M^{2}=I$
(D) $(\text{adj} M)^{2}=I$
(C) $M^{2}=I$
(D) $(\text{adj} M)^{2}=I$
Was this answer helpful?
0
0
Top JEE Advanced Mathematics Questions
- In a $\triangle$ ABC w ith fixed base BC, the vertex A moves such that cos B + cos C = 4 $ sin^2 \frac{A}{2}$. If a, b and c denote th e lengths of th e sides of th e triangle opposite to the angles A, B and C respectively, then
- Let $P=[a_{ij}]$ be a $3\times3$ matrix and let $Q=[b_{ij}]$ where $b_{ij}=2^{i+j} a_{ij}$ for $1 \le i, j \le $.If the determinant of $P$ is $2$, then the determinant of the matrix $Q$ is
- For the equation $3x^2+px+3=0,p>0,$ if one of the root is square of the other, then p is equal to
- The equation $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}$ has
Let \(f(x)=x+log_{e}x−xlog_{e}x,\text{ }x∈(0,∞)\).
- Column 1 contains information about zeros of \(f(x)\), \(f'(x)\) and \(f''(x)\).
- Column 2 contains information about the limiting behavior of \(f(x)\), \(f'(x)\) and \(f''(x)\) at infinity.
- Column 3 contains information about increasing/decreasing nature of \(f(x)\) and \(f'(x)\).
View More Questions
Top JEE Advanced Applications of Determinants and Matrices Questions
- The parameter on which the value of the determinant $\Rightarrow \Delta=\begin{vmatrix}1& a & a^2\\ \cos(p-d)x& \cos\ px & \cos(p+d)x\\ \sin(p-d)x& \sin\ px& \sin(p+d)x\\ \end{vmatrix}$ does not depend upon, is
- Let α, β and γ be real numbers such that the system of linear equations
\(x + 2y + 3z = α\)
\(4x + 5y + 6z = β\)
\(7x + 8y + 9z = γ – 1 \)
is consistent. Let\( |M|\) represent the determinant of the matrix.
\(M = \begin{bmatrix} \alpha & 2 & \gamma &\\ \beta & 1 & 0 & \\-1& 0&1 \end{bmatrix}\)
Let P be the plane containing all those \((α, β, γ)\) for which the above system of linear equations is consistent, and D be the square of the distance of the point (0, 1, 0) from the plane P.
Top JEE Advanced Questions
- Yellow light is used in a single slit diffraction experiment with slit width of 0.6 mm. If yellow light is replaced by X-rays, then the observed pattern will reveal
- The size of the image of an object, which is at infinity, as formed by a convex lens of focal length 30 cm is 2 cm. If a concave lens of focal length 20 cm is placed between the convex lens and the image at a distance of 26 cm from the convex lens, calculate the new size of the image.
- A concave lens of glass, refractive index 1.5 has both surfaces of same radius of curvature R. On immersion in a medium of refractive index 1.75, it will behave as a
- The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation $\vec{r}(t)=\alpha t^{3} \hat{i}+\beta t^{2} \hat{j}$ where $\alpha=\frac{10}{3} \,ms ^{-3}, \beta=5\, ms ^{-2}$ and $m =0.1 \,kg$. At $t =1\, s$, which of the following statement (s) is (are) true about the particle?
- In an n-p-n transistor circuit, the collector current is 10 mA. If 90% of the electrons emitted reach the collector
View More Questions