Let A be a $2 \times 2$ matrix with det $( A )=-1$ and $\operatorname{det}(( A + I )(\operatorname{Adj}( A )+ I ))=4$. Then the sum of the diagonal elements of A can be:
- $-1$
- 2
- 1
- $-\sqrt{2}$
The Correct Option is B
Solution and Explanation
The correct option is (B) : 2
Given : Relation det ((A + I)(adj(A) + I)) = 4 , det (A) = -1,
Then, adj A = -A-1
| (A + I )A-1 + I | = 4
| -I + A - A-1 + I | =4
| A - A-1 | = 4
Let A \(=\begin{bmatrix} a & b \\ c & d \end{bmatrix} \) then A-1 = \(\begin{bmatrix} -d & b \\ c & -a \end{bmatrix}\)
| A - A-1 | = \(\begin{bmatrix} a+d & 0 \\ 0 & d+a \end{bmatrix}=4\)
(a + d)2 = 4
⇒ a + d = ± 2
⇒ | a + d | = 2
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.