Problem 179 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A=\left[\begin{array}{cc}2 & -1 \\ 0 & 1\end{array}\right]\) and \(B=\left[\begin{array}{cc}1 & 0 \\ -1 & -1\end{array}\right]\), show that \((A+B)^{2}=A^{2}+A B+B A+B^{2} \neq A^{2}+2 A B+B^{2}\)

Step-by-Step Solution

Answer

Adding the two given matrices A and B and then squaring the result gives a different matrix compared to squaring the matrices A and B individually, adding the results to the matrix resulting from multiplying A and B twice in different order. So, \( (A+B)^2 \ne A^2+2AB+B^2 \), proving that matrix multiplication is not commutative.

Step 1: Compute A+B

Add the matrices A and B together element by element. Apply matrix addition, which is simply the addition of corresponding elements: \( C_{ij} = A_{ij} + B_{ij} \) where \( A_{ij} \) and \( B_{ij} \) are elements of matrices A and B respectively. This results in a new matrix C where \( C = A + B \)

Step 2: Compute (A+B)^2

To compute \( (A+B)^2 \), you multiply matrix \( (A+B) \) by itself. Matrix multiplication requires sum of the products of corresponding elements from the rows of the first matrix and the columns of the second. The result is a new matrix.

Step 3: Compute A^2, AB, BA and B^2

Compute these four elements. For A^2 and B^2, multiply each matrix by itself. For AB, multiply matrix A by matrix B. For BA, multiply matrix B by matrix A. It's crucial to remember that in matrix multiplication, the order of multiplication matters.

Step 4: Compare Results

We then compare the matrix we got in step 2 \( (A+B)^2 \) with the sum of the matrices we got in step 3. This is how we demonstrate the inequality \( (A+B)^2 \ne A^2+2AB+B^2 \) proving that matrix multiplication is not commutative.

Key Concepts

Matrix Addition

Matrix addition is a process similar to how we add numbers, only here we add the corresponding elements of two matrices. For instance, if we have two matrices,

• \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \)
• \( B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \)

Matrix addition involves adding each element from matrix A to the corresponding element in matrix B.
The result is a new matrix:

• \( C = A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} \)

This concept is very intuitive. However, when adding matrices, they must have the same dimensions, meaning the number of rows and columns must be equal in both matrices.

Matrix Multiplication

Matrix multiplication is a bit more involved than addition. It involves a row-by-column operation. To multiply two matrices,

• lets say \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \)
• and \( B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \)

You take the elements of the rows from the first matrix and the columns from the second matrix.
The result is the matrix \( C \), where each element \( c_{ij} \) is calculated as follows:

• \( c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} \)

It is important to note that for matrix multiplication to be possible, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This makes matrix multiplication more complex but allows for diverse operations and applications in different mathematical and real-world contexts.

Non-Commutative Property

In matrix algebra, the non-commutative property is crucial to understand. Unlike normal arithmetic—where, for example, multiplication is commutative (i.e., \( a \times b = b \times a \))—matrix multiplication does not share this property.
If we have two matrices A and B, the product \( AB \) is typically different from \( BA \). This signifies that the order in which you multiply matrices affects the result.

• In simpler terms, changing the order will lead often lead to different outcomes—\( AB eq BA \)

This property holds significant mathematical implications, particularly in understanding the behavior of various mathematical problems and their real-world applications. In the exercise example, this property is highlighted when comparing \( AB \) with \( BA \). This distinction makes matrix algebra unique and versatile.

Matrix Square

Squaring a matrix entails multiplying the matrix by itself. If you have a matrix \( A \), computing \( A^2 \) means performing the operation \( A \times A \).
This involves matrix multiplication, and you follow the same process where each element \( c_{ij} \) is based on the multiplication of corresponding row and column elements.

• For example, if \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), then \( A^2 = A \times A = \begin{bmatrix} a^2 + bc & ab + bd \ ac + cd & bc + d^2 \end{bmatrix} \)

Matrix squaring is useful in various mathematical computations and transformations. Understanding how to square a matrix can aid in solving complex algebraic problems and demonstrates how combinations of matrix operations can be applied across different dimensional contexts.