Problem 180 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A=\left[\begin{array}{cc}-1 & 2 \\ 2 & 3\end{array}\right], B=\left[\begin{array}{ll}3 & 0 \\ 1 & 1\end{array}\right]\). Verify 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

After calculating separately we get, \((A+B)^{2}=A^{2}+A B+B A+B^{2} = \left[\begin{array}{cc}10 & 14 \ 18 & 22\end{array}\right] \neq A^{2}+2 A B+B^{2}\) which means the equality and inequality both hold true.

Step 1: Calculate A+B

Sum the individual entries of matrices \(A\) and \(B\) i.e. \(A+B = \left[\begin{array}{cc}-1 & 2 \ 2 & 3\end{array}\right] + \left[\begin{array}{cc}3 & 0 \ 1 & 1\end{array}\right] = \left[\begin{array}{cc}2 & 2 \ 3 & 4\end{array}\right]\)

Step 2: Calculate (A+B)²

Multiply the matrix \(A+B\) by itself i.e. \((A+B)^2 = \left[\begin{array}{cc}2 & 2 \ 3 & 4\end{array}\right] * \left[\begin{array}{cc}2 & 2 \ 3 & 4\end{array}\right] = \left[\begin{array}{cc}10 & 14 \ 18 & 22\end{array}\right]\)

Step 3: Calculate A², AB, BA, B² and verify the expression

Next, calculate separately \(A^2, AB, BA,\) and \(B^2\) and sum the matrices as per the given expression. If the total equals the result from Step 2, the expression holds.

Step 4: Calculate and compare with the second expression

Similarly, calculate and sum \(A^2, 2AB, B^2\) and if this total is different from the result in Step 2, then the given inequality also holds.

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. Unlike simple arithmetic with numbers, multiplying matrices involves a process that combines the rows of the first matrix with the columns of the second.
For two matrices \( A \) and \( B \) to be multiplied, the number of columns in \( A \) must be equal to the number of rows in \( B \). The resulting product matrix has dimensions derived from the rows of \( A \) and the columns of \( B \).
Here’s how you multiply matrices:

• Take each row element of the first matrix \( A \).
• Multiply by the corresponding column element of the second matrix \( B \).
• Sum all the products for a single position in the resulting matrix.

This concept is crucial in step 2 and step 3 of the solution where we calculated \((A+B)^2\), and individual products like \(A^2\), \(AB\), \(BA\), and \(B^2\). Each of these requires its multiplication process.

Matrix Addition

Matrix addition is an operation where you add corresponding elements of two matrices of the same dimension. It’s like doing arithmetic with pairs of numbers, only this time, the numbers are arranged in rows and columns within a matrix.
To perform matrix addition:

• You go through each element, row by row.
• Add the first matrix's element to its corresponding one in the second matrix.
• The result is a new matrix with these sums as each element.

The process was used in Step 1 of the problem where we summed matrices \(A\) and \(B\), resulting in a new matrix \(A + B\) with elements derived from adding each respective element of \(A\) and \(B\). Understanding this process is key, as it prepares the matrix for later multiplication in the following steps.

Inequalities in Matrices

Inequalities in matrices refer to comparing not the individual elements, but the resulting matrices from algebraic expressions and verifying their equality or inequality.
In our exercise, we encounter two expressions that appear similar but reveal an inequality when calculated. These expressions, \((A+B)^2 = A^2 + AB + BA + B^2 \) and \((A+B)^2 eq A^2 + 2AB + B^2 \), demonstrate non-commutative properties due to the order of multiplication.
Here’s why inequalities arise:

• The ordering of multiplication is critical. Unlike numbers, you cannot rearrange matrices arbitrarily.
• Matrix multiplication is non-commutative, meaning \(AB eq BA\).

These principles ensure while individual components of matrices combine in various ways, the overall expressions can diverge significantly in results. The inequality highlights that linear transformation applications can't always be simplified as with traditional arithmetic.