Problem 172 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right], B=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\), then find \(A B\) and \(B A\) and show that \(A B \neq B A\).

Step-by-Step Solution

Answer

After performing the required operations, the matrices \(AB =\) \(\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}\) and \(BA =\) \(\begin{bmatrix}0 & 1 \\ 1 & 1\end{bmatrix}\) are obtained. This demonstrates that in the case of these particular matrices A and B, it is true that \(AB \neq BA\).

Step 1: Matrix A multiplied by Matrix B

The multiplication of two matrices is carried out element by element. Each element of the result matrix corresponds to the sum of the products of corresponding elements from the multiplied matrices. Therefore, the multiplication of A and B (denoted as AB) is calculated as follows: \(AB =\) \(\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}\cdot \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}= \) \(\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}\).

Step 2: Matrix B multiplied by Matrix A

In order to find the product of B and A (BA), the process from step 1 is repeated, but this time B is multiplied by A: \(BA = \) \(\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \cdot \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}= \) \(\begin{bmatrix}0 & 1 \\ 1 & 1\end{bmatrix} \).

Step 3: Verify that AB is not equal to BA

From step 1 and step 2, we see that the results of AB and BA are not equal, thus proving that \(AB \neq BA\). Here is a direct comparison for clarity: \(AB =\) \(\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}\) while \(BA =\) \(\begin{bmatrix}0 & 1 \\ 1 & 1\end{bmatrix}\). It can be observed that the matrices are indeed different.

Key Concepts

Non-Commutative Property

In the realm of matrices, understanding the non-commutative property is crucial. This property asserts that the order in which you multiply matrices matters. Typically, we are used to numbers, which follow the commutative property in multiplication, such as: \( a \times b = b \times a \). However, matrices behave differently.
This non-commutative nature can be demonstrated with an example given matrices \(A\) and \(B\):

• When you multiply \(A\) by \(B\), you might end up with a different matrix than when you multiply \(B\) by \(A\).
• In the exercise, \(AB = \begin{bmatrix}1 & 1 \ 0 & 1\end{bmatrix}\) while \(BA = \begin{bmatrix}0 & 1 \ 1 & 1\end{bmatrix}\), showing a clear difference.

This means that swapping the matrices changes the result. This property is essential when performing calculations in matrix algebra, as it affects how equations with matrices are solved. Always double-check the order of multiplication in your calculations.

Matrix Algebra

Matrix algebra is a sophisticated branch of mathematics that deals with matrices, which are collections of numbers arranged in rows and columns. Understanding its operations is key for solving various mathematical and real-world problems.

When we talk about matrix algebra, several operations come into play:

• Addition: You can add two matrices by adding their corresponding elements. However, they must be of the same dimension.
• Subtraction: Similar to addition, subtract by taking difference of corresponding elements in matrices of the same size.
• Multiplication: When multiplying matrices, row elements of the first must be multiplied with column elements of the second matrix. The process involves summing these products to form elements of the resulting matrix.

Crucially, the non-commutative property discussed earlier is a part of this. It reminds us that while adding matrices is straightforward, multiplying them demands attention to detail. Through practice, these operations become intuitive, empowering you to tackle complex mathematical models with matrix algebra.

Elementary Matrices

Elementary matrices are simple and essential matrices used primarily to execute transformations and solve linear equations. The magic of elementary matrices lies in their ability to represent basic row operations like swapping rows, scaling rows, or adding multiples of rows to each other.

Each elementary matrix arises from an identity matrix, which is a square matrix with ones on the diagonal and zeros elsewhere. You can create an elementary matrix by performing a single row operation on the identity matrix directly.

• Row Switching: An elementary matrix that swaps two rows can reverse their order in another matrix.
• Scaling: Scalar multiplication of a row is achieved by multiplying an entire row with a non-zero scalar value.
• Additive Combination: By adding a multiple of one row to another, an elementary matrix applies this operation to another matrix.

In linear algebra, they are the building blocks for more complicated operations. When multiplying any matrix by its corresponding elementary matrix, you perform that specific row operation on the matrix. This makes them foundational in matrix methods for solving simultaneous equations, finding inverses, or simplifying matrices.