Let the matrix A = begin{bmatrix} 10^{30} + 5 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) To determine if $A$ is invertible,we check if the determinant $|A| 
eq 0$. We consider the parity of the elements in the matrix by taking the mat

Vedclass · Accepted Answer

$A$ is invertible for all $n \in N$

Vedclass · Answer

(A) To determine if $A$ is invertible,we check if the determinant $|A| 
eq 0$. We consider the parity of the elements in the matrix by taking the matrix modulo $2$. Modulo $2$,the matrix $A$ becomes: $A \equiv \begin{bmatrix} 0+1 & 0+0 & 0+0 \ 0+0 & 0+1 & 0+0 \ 0+0 & 0+0 & 0+1 \end{bmatrix} \pmod{2}$ $A \equiv \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \pmod{2}$ The determinant $|A| \pmod{2}$ is the determinant of the identity matrix,which is $1$. Since $|A| \equiv 1 \pmod{2}$,the determinant $|A|$ must be an odd number. An odd number cannot be $0$. Therefore,$|A| 
eq 0$ for all $n \in N$. Thus,the matrix $A$ is invertible for all $n \in N$.

Let the matrix $A = \begin{bmatrix} 10^{30} + 5 & 10^{20} + 4 & 10^{20} + 6 \\ 10^4 + 2 & 10^8 + 7 & 10^{10} + 2n \\ 10^4 + 8 & 10^6 + 4 & 10^{15} + 9 \end{bmatrix}$,where $n \in N$. Then:

Similar Questions

Let $A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix}$. Verify that $A(\text{adj } A) = (\text{adj } A) A = |A| I$.

Find the inverse of the matrix,if it exists: $\left[\begin{array}{cc}6 & -3 \\ -2 & 1\end{array}\right]$

For the matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix}$,show that $A^{3} - 6A^{2} + 5A + 11I = 0$. Hence,find $A^{-1}$.

If $A$ and $B$ are non-singular matrices,then which of the following is true?

If $A$ is an invertible matrix of order $n$, then the determinant of $\operatorname{adj} A$ is equal to :

Vedclass Test Series

Exam Paper Generator

Online Exam Module