If A and B are two n times n non-singular matrices, then
If A and B are two n times n non-singular matrices, then
Show Hint
- AB is non-singular
- AB is singular
- (AB)^-1 = A^-1 B^-1
- (AB)^-1 does not exist
The Correct Option is A
Solution and Explanation
Step 1: Understanding non-singular matrices.
A non-singular matrix is a square matrix that has an inverse. The product of two non-singular matrices is always non-singular.
Step 2: Analyzing the product of two matrices.
If both A and B are non-singular matrices, then their product AB is also non-singular. This is because the determinant of AB is the product of the determinants of A and B, and the determinant of a non-singular matrix is non-zero.
Step 3: Exploring other options.
(A) AB is non-singular: Correct. The product of two non-singular matrices is non-singular.
(B) AB is singular: Incorrect. This is not true if both matrices are non-singular.
(C) (AB)^-1 = A^-1 B^-1 : Incorrect. The correct inverse of AB is (AB)^-1 = B^-1 A^-1 , not A^-1 B^-1 .
(D) (AB)^-1 does not exist: Incorrect. The inverse of AB exists, as long as A and B are non-singular.
Step 4: Conclusion.
Thus, the correct answer is (A) AB is non-singular.
Final Answer: AB is non-singular.
Top Questions on Matrices
- 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 $A^{T}A$ is $5$, is
- For the matrices \( A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \) and \( B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} \), if \( (A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix} \), then among the following which one is true?}
- Let matrix \[ A=\begin{pmatrix} 3 & -4\\ 1 & -1 \end{pmatrix} \] and \(A^{100} = 100B + I\). Find the sum of all the elements in \(B^{100}\).
- If \[ X=\begin{bmatrix}x\\y\\z\end{bmatrix} \] is a solution of the system of equations $AX=B$, where \[ \text{adj }A= \begin{bmatrix} 4 & 2 & 2\\ -5 & 0 & 5\\ 1 & -2 & 3 \end{bmatrix} \quad \text{and} \quad B=\begin{bmatrix}4\\0\\2\end{bmatrix}, \] then $|x+y+z|$ is equal to
Let \[ f(x)=\int \frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}\,dx \] and $f(1)=\frac14$. Given that
and $B=\operatorname{adj}(\operatorname{adj}A)$, if $|B|=81$, find the value of $\alpha^2$ (where $\alpha\in\mathbb{R}$).
Questions Asked in JKBOSE XII exam
- When two coins are tossed, what is the probability of at most two heads?
- JKBOSE XII - 2026
- Statistics and Probability
- Find the derivative of tan⁻¹(x) + cos⁻¹(x)
- JKBOSE XII - 2026
- Derivatives
- Find ∫ dx / (sin²x cos²x)
- JKBOSE XII - 2026
- Integration
- The differential equation of all circles of radius a is of order
- JKBOSE XII - 2026
- Differential equations
a times b is equal to
- JKBOSE XII - 2026
- Geometry and Vectors