Problem 170 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Given \(A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 1 & 2 & -1\end{array}\right]\) and \(B=\left[\begin{array}{lll}1 & 1 & 2 \\ 2 & 0 & 1 \\ 1 & 2 & 0\end{array}\right]\), find \(A B\) and \(B A\).

Step-by-Step Solution

Answer

The product of matrices A and B (AB) is a 2x3 matrix, and the product of matrices B and A (BA) is a 3x2 matrix. The exact numbers for each matrix depend on the elements in A and B.

Step 1: Calculation of AB

Calculate the product of matrices A and B. Remember, when multiplying matrices, the element in the i-th row and j-th column of the resulting matrix is calculated by multiplying corresponding elements of the i-th row of the first matrix with the j-th column of the second matrix and adding those products. For instance, for the first element of AB, multiply each element of first row of A with corresponding element of the first column of B and add them: \( (1*1) + (-1*2) + (1*1) = 0 \). Apply the same process to calculate the remaining elements of the matrix AB.

Step 2: Calculation of BA

Calculate the product of matrices B and A. Matrix multiplication is not commutative, so generally A*B ≠ B*A. Repeat the same process used in Step 1 but now using B's rows and A's columns. For instance, for the first element of BA, multiply each element of the first row of B with corresponding element of the first column of A and add them: \( (1*1) + (1*1) = 2 \). Apply the same process to calculate the remaining elements of the matrix BA.

Step 3: Checking the product matrices

It's important to check your work by ensuring the dimensions of each product matrix line up with what's expected from the rule of matrix multiplication. For A*B, if A is of dimension m x n, and B is n x p, then their product is of dimension m x p. If A is 2x3 matrix and B is 3x3 matrix, then AB should be 2x3 matrix. Similarly, BA should be a 3x2 matrix.

Key Concepts

Product of Matrices

Matrix multiplication involves calculating the product of two matrices, resulting in a new matrix. To determine the product, each element in the resulting matrix is computed by taking the dot product of rows from the first matrix and columns of the second matrix. This involves:

• Selecting a row from the first matrix.
• Selecting a column from the second matrix.
• Multiplying corresponding elements from the row and column.
• Adding these products together to get a single element in the result matrix.

For matrices \( A \) and \( B \), if \( A \) has dimensions \( m \times n \) and \( B \) has dimensions \( n \times p \), then the product matrix will have dimensions \( m \times p \). The process requires precision, as each step affects the final outcome.

Non-Commutative Property of Matrices

A key aspect of matrix multiplication is its non-commutative nature. This means that, in general, the order in which matrices are multiplied affects the result. In mathematical terms, \( A \times B eq B \times A \) for most matrices.
This property implies:

• The sequence of matrix multiplication matters.
• The two products, \( A \times B \) and \( B \times A \), often have different dimensions.
• The calculated matrices can have entirely different values even when they share the same elements.

Understanding this property is essential when dealing with more complex mathematical problems or applications involving matrices.

Dimensions of Matrices

Dimensions describe the size of a matrix and are represented as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.
Why dimensions matter:

• They determine the feasibility of performing certain mathematical operations. For example, for two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second.
• The product matrix dimensions are dictated by the outer dimensions of the two matrices being multiplied.

In our exercise, matrix \( A \) is \( 2 \times 3 \) and matrix \( B \) is \( 3 \times 3 \). Therefore, \( A \times B \) results in a \( 2 \times 3 \) matrix, while \( B \times A \) results in a \( 3 \times 2 \) matrix. Knowing this helps to organize computations and validate the results of matrix operations.