Problem 174 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\), then find \(A^{2}\) and \(A^{3}\).

Step-by-Step Solution

Answer

The square of A, or \(A^{2}\), is \(\left[\begin{array}{cc}8 & 5 \ -2 & 3\end{array}\right]\) and the cube of A, or \(A^{3}\), is \(\left[\begin{array}{cc}18 & 29 \ -23 & -2\end{array}\right]\).

Step 1: Find \(A^{2}\)

First find \(A^{2}\) or \(A*A\) using matrix multiplication. The element at the i-th row and j-th column in the resulting matrix is the sum of the product of elements from the i-th row of the first matrix and the j-th column of the second one. For instance, the element at the 1st row and 1st column is 3*3 + 1*(-1) = 8.

Step 2: Continue matrix multiplication for \(A^{2}\)

Continue the multiplication process for the other elements. As a result, the matrix \(A^{2}\) would be \(\left[\begin{array}{cc}8 & 5 \ -2 & 3\end{array}\right]\).

Step 3: Find \(A^{3}\)

Find \(A^{3}\) or \(A^{2} * A\) using similar steps as above. Here, \(A^{2}\) is the resulting matrix obtained above and A is the original matrix. As a result, the matrix \(A^{3}\) would be \(\left[\begin{array}{cc}18 & 29 \ -23 & -2\end{array}\right]\).

Key Concepts

Squared Matrices

A squared matrix, simply put, is the result of multiplying a matrix by itself. For a matrix \( A \), squaring it involves performing what is known as matrix multiplication between \( A \) and itself.

To find \( A^2 \), you follow a methodical process:

• Take each row of the first matrix \( A \) and each column of the second matrix \( A \). For instance, if \( A = \begin{bmatrix} 3 & 1 \ -1 & 2 \end{bmatrix} \).
• For each element in the product matrix, multiply corresponding elements from the row and the column and add those products together. For example, the element at the first row and first column of \( A^2 \) would be calculated as: \( 3 \times 3 + 1 \times (-1) = 8 \).

This straightforward method is key to understanding and computing squared matrices effectively.

Cubed Matrices

Cubing a matrix involves multiplying it by itself twice. In mathematical terms, for a matrix \( A \), \( A^3 \) refers to \( A^2 \times A \).

Here's how you can find \( A^3 \):

• Start with \( A^2 \), the squared matrix you previously calculated. If \( A^2 = \begin{bmatrix} 8 & 5 \ -2 & 3 \end{bmatrix} \), then you use this as the starting point.
• Next, perform matrix multiplication of \( A^2 \) with the original matrix \( A \). The process involves taking rows from \( A^2 \) and columns from \( A \), multiplying their elements, and summing the products.
• For example, the element in the first row, first column of \( A^3 \) is derived from: \( 8 \times 3 + 5 \times (-1) = 18 \).

By following this systematic approach, you accurately find the cubed matrix result \( \begin{bmatrix} 18 & 29 \ -23 & -2 \end{bmatrix} \).

Matrix Operations

Matrix operations are fundamental in mathematics and involve a series of calculations for manipulating matrices. Common operations include addition, subtraction, and multiplication. But here, let's focus on multiplication, especially as it relates to squaring and cubing matrices.

Matrix multiplication is not always commutative, meaning \( A \times B \) usually does not equal \( B \times A \). To multiply matrices:

• Ensure the number of columns in the first matrix equals the number of rows in the second. This compatibility is essential for multiplication.
• Use the formula: for matrix \( C = AB \), each element \( c_{ij} \) in \( C \) is calculated as the sum of products of corresponding elements from row \( i \) in \( A \) and column \( j \) in \( B \).

Understanding these rules helps when performing advanced operations such as finding \( A^2 \) or \( A^3 \). These operations are key to exploring more complex algebraic structures and transformations.