Let \( S = \{ M = [a_{ij}], a_{ij} \in \{0, 1, 2\}, 1 \leq i, j \leq 2 \} \) be a sample space and \( A = \{ M \in S : M \text{ is invertible} \} \) be an event. Then \( P(A) \) is equal to:
- \(\frac{16}{27}\)
- \(\frac{50}{81}\)
- \(\frac{47}{81}\)
- \(\frac{46}{81}\)
The Correct Option is B
Solution and Explanation
Given \( M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), where \( a, b, c, d \in \{0, 1, 2\} \). The number of elements in the sample space \( S \) is \( 3^4 = 81 \). For \( M \) to be invertible, its determinant must be non-zero: \[ \text{det}(M) = ad - bc \neq 0 \] We compute the valid combinations for which \( ad - bc \neq 0 \). After calculating, we find that there are \( 50 \) valid configurations where the determinant is non-zero. Thus, the probability \( P(A) = \frac{50}{81} \).
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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.