Question

If A and B are two matrices such that AB is an identity matrix and the order of matrix B is \(3 \times 4\), then the order of matrix A is:

• A. \(3 \times 3\)
• B. \(4 \times 3\)
• C. \(4 \times 4\)
• D. \(3 \times 4\)

Hint:

In matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

Answer 1

The Correct Option is B

If \( A \) and \( B \) are two matrices such that \( AB \) is an identity matrix and the order of matrix \( B \) is \( 3 \times 4 \), then the order of matrix \( A \) is:

Step 1: Properties of matrix multiplication
We are given that \( AB = I \), where \( I \) is the identity matrix. For matrix multiplication to be valid, the number of columns in matrix \( A \) must equal the number of rows in matrix \( B \). Therefore, if the order of matrix \( B \) is \( 3 \times 4 \), matrix \( A \) must have 4 rows.

Step 2: The order of matrix \( A \)
Since \( AB = I \), the resulting matrix is the identity matrix. The identity matrix \( I \) has the same order as the number of rows of matrix \( A \) and the number of columns of matrix \( B \). The identity matrix for matrices of this type will have the order \( 3 \times 3 \), since matrix \( B \) has 4 columns, and for the product to result in a square matrix (the identity matrix), matrix \( A \) must have 3 columns. Therefore, matrix \( A \) must have 4 rows and 3 columns.

Step 3: Conclusion
The order of matrix \( A \) is \( 4 \times 3 \).