Let A be a \(2×2\) matrix with \(det (A) = –1\) and \(det((A + I) (Adj (A) + I)) = 4\). Then the sum of the diagonal elements of A can be
\(-1\)
\(2\)
\(1\)
\(-\sqrt 2\)
The Correct Option is B
Solution and Explanation
\(|(A+I)(adj\ A+I)|=4\)
\(⇒|A\ adj\ A +A+adj \ A+I|=4\)
\(⇒|(A)I+A+adj\ A+I|=4\)
\(|A|=−1\)
\(⇒|A+adj\ A|=4\)
\(A=\begin{bmatrix} a & b\\[0.3em] c & d \\[0.3em] \end{bmatrix}\)
\(adj A=\begin{bmatrix} a & -b\\[0.3em] -c & d \\[0.3em] \end{bmatrix}\)
\(⇒ \begin{bmatrix} (a+d) & b\\[0.3em] 0 & (a+d) \\[0.3em] \end{bmatrix}=4\)
\(⇒ a + d = ±2\)
So, the correct option is (B): \(2\)
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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.