Problem 190 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\), then find \(A^{11}\).

Step-by-Step Solution

Answer

The 11th power of matrix \( A \), \( A^{11} = A \).

Step 1: Observe the Given Matrix

First look at the given matrix \( A = \left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}\right] \). Note that interchanging the numbers in the rows or the columns results in the same matrix. Now, let's see what happens when the matrix is squared.

Step 2: Square the Given Matrix

Calculate the square of the given matrix to see the pattern it forms. To square a matrix, the matrix is multiplied by itself as follows: \( A^2 = A \cdot A = \left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}\right]\cdot\left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}\right] = \left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right] \). When calculated, we obtain the identity matrix. This leads us to the conclusion that every even power of the matrix \( A \) will result in the identity matrix.

Step 3: Determine the Power of the Matrix

Now, to find \( A^{11} \), we can use the property that every even power of \( A \) equals the identity matrix. The number 11 can be written as 10 + 1, so we can rewrite \( A^{11} \) as \( A^{10}A^{1} = I \cdot A =A \). Hence, \( A^{11} = A \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying rows by columns to produce a new matrix. When multiplying two matrices, every element of the product matrix is derived from the sum of the products of rows and columns of the two matrices involved. Let's break it down more simply:

• Each element in the resulting matrix is calculated by taking the dot product of corresponding rows and columns from the matrices you are multiplying.
• This operation is not commutative, meaning that in general, \( AB eq BA \).
• The number of columns in the first matrix must equal the number of rows in the second matrix.

Consider two matrices \( A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \), to multiply them: we take one row from \( A \) and one column from \( B \), each of which will create one element in the new matrix. This product results in the identity matrix following the calculation of \( A^2= A \cdot A \).
Once you grasp this concept, it becomes quite clear why pattern recognition in matrix multiplication can greatly save time with operations such as determining higher powers of matrices.

Identity Matrix

The identity matrix, often denoted by \( I \), is a special form of matrix with ones on the diagonal and zeros elsewhere. Think of it like the number 1 in matrix operations—not altering any matrix it multiplies, effectively acting as the multiplicative identity.

• If \( I \) is multiplied by any matrix \( A \), the result is \( A \), symbolically written as \( AI = IA = A \).
• For a 2x2 matrix, the identity matrix looks like \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).

In our exercise, it's observed that every even power of matrix \( A \) results in the identity matrix. This means squaring \( A \) or taking any even power yields \( I \), essentially outputting the identity matrix, e.g., \( A^2 = I \). In essence, multiplying by the identity matrix gives a result that doesn’t change the multiplied matrix, a useful feature when applying algebraic operations on matrices.

Matrix Patterns

Matrix patterns refer to the recognitions and repetitions within matrix operations that simplify computational processes. Observing these patterns can make predicting future computations easier without actually performing all the operations.
Take the example in the exercise. By observing the pattern when calculating matrix powers of \( A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \), we find:

• \( A^1 \) results in the original matrix \( A \).
• \( A^2 \) results in the identity matrix \( I \).

From one cycle (two consecutive powers here), it becomes clear that odd powers of \( A \) give \( A \) itself, while even powers give \( I \), the identity matrix. This discovery allows one to predict future powers of \( A \) effortlessly by identifying and utilizing the pattern repeatedly; for example, \( A^{11} = A^{10} \cdot A = I \cdot A = A \). Recognizing such patterns in matrix operations can vastly reduce computational efforts in larger or more complex problems.