Problem 195 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\) and \(A^{2}=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]\), then show that \(\alpha=a^{2}+b^{2}, \beta=2 a b\).

Step-by-Step Solution

Answer

The given expressions \(\alpha=a^{2}+b^{2}\) and \(\beta=2ab\) are true as they represent the elements of the squared matrix \(A^{2}\) derived from the original matrix \(A\).

Step 1: Square the Matrix A

In order to find \(A^{2}\), you need to multiply the matrix \(A\) by itself. This is done by the rule of matrix multiplication: \[A^{2} = A \times A = \left[\begin{array}{ll}a & b \ b & a\end{array}\right] \times \left[\begin{array}{ll}a & b \ b & a\end{array}\right] = \left[\begin{array}{ll}a^{2}+b^{2} & ab + ba \ ab + ba & a^{2}+ b^{2}\end{array}\right]\]

Step 2: Compare Elements of Resulting Matrix with Matrix \(A^{2}\)

In matrix \(A^{2}\), we have the elements \(\alpha\) and \(\beta\) defined as \(\alpha = a^{2}+b^{2}\) and \(\beta = ab + ba = 2ab\) respectively when we compare it to the result we obtained in Step 1 resulting from squaring matrix \(A\). As you see in our matrix multiplication, the elements in the resulting matrix are the same as those in matrix \(A^{2}\). Therefore, \(\alpha=a^{2}+b^{2}\) and \(\beta=2ab\).

Step 3: Conclusion

You have successfully shown that \(\alpha=a^{2}+b^{2}\) and \(\beta=2ab\) as required by the problem statement. These results show the relationship between elements of the original matrix \(A\) and the squared matrix \(A^{2}\).

Key Concepts

Symmetric Matrices

A symmetric matrix is a special kind of matrix. These matrices are equal to their own transposes. What does that mean? Simply put, the elements are mirrored along the main diagonal. That means, if you swap rows and columns, the matrix remains unchanged.
For example, if matrix \( A = \left[\begin{array}{cc} a & b \ b & a \end{array}\right] \) is symmetric, you will find that the entries above the main diagonal are the same as those below it.

• Entry (1, 2) is equal to Entry (2, 1).
• The diagonal entries are the same for both rows and columns.

Symmetric matrices often arise in various applications, such as in solving systems of equations, physics, and statistics, because of their simplicity and robustness.
Understanding symmetric properties helps in simplifying problems since their structure reduces the amount of information you need to process.

Matrix Squaring

Squaring a matrix involves multiplying the matrix by itself. This is similar to squaring a number, but matrix multiplication is a bit more involved.
To square a matrix, follow these steps:

• Multiply each element of the first row by each corresponding element in the first column.
• Continue this pattern for all rows and columns.
• Sum the products for each position to get the new element in the resultant matrix.

Consider matrix \( A = \left[\begin{array}{cc} a & b \ b & a \end{array}\right] \). When you multiply it by itself:\[ A^2 = \left[\begin{array}{cc} a & b \ b & a \end{array}\right] \times \left[\begin{array}{cc} a & b \ b & a \end{array}\right] = \left[\begin{array}{cc} a^2 + b^2 & 2ab \ 2ab & a^2 + b^2 \end{array}\right] \]
Squaring simplicity arises because of the repeated values in symmetric matrices, allowing for quick computation.

Element-Wise Comparison

Element-wise comparison is crucial for verifying the results of matrix operations. It involves checking each corresponding element of two matrices to see if they match.
In our problem, we derived a new matrix \( A^2 \) by squaring matrix \( A \).
Then, we compared it to the matrix \( \left[\begin{array}{cc} \alpha & \beta \ \beta & \alpha \end{array}\right] \).
For successful comparison:

• Each \( \alpha \) should equal \( a^2 + b^2 \).
• Each \( \beta \) should equal \( 2ab \).

In this problem, confirming each element matches ensures that our original equation holds true.
This process not only verifies calculations but also boosts confidence in understanding properties like matrix symmetry and multiplication.