For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $I$ be the $3 \times 3$ identity matrix Let $E$ and $F$ be two $3 \times 3$ matrices such that $( I - EF )$ is invertible. If $G =( I - EF )^{-1}$, then which of the following statements is(are) TRUE?
- $| FE |=| I - FE || FGE |$
- $( I - FE )( I + FGE )= I$
- $EFG = GEF$
- $( I - FE )( I - FGE )= I$
The Correct Option is A, B, C
Solution and Explanation
\(I – EF = G^{–1}\)
\(G – GEF = I …(1)\)
And \(G – EFG = I …(2)\)
Clearly \(GEF = EFG\) (option C is correct)
Also \((I – FE)(I + FGE) = I – FE + FGE – FE + FGE\)
\(= I – FE + FGE – F(G – I)E\)
\(= I – FE + FGE – FGE + FE\)
\(= I\) (option B is correct and D is incorrect)
Now, \((I – FE)(I – FGE) = I – FE – FGE + F(G – I)E\)
\(= I – 2FE\)
\((I – FE)(- FGE) = – FE\)
\(|I – FE||FGE| = |FE|\)
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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.