Problem 186 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Show that \(E^{2} F+F^{2} E=E\), where \(E=\left[\begin{array}{lll}0 & 0 & 1 \\\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right], F=\left[\begin{array}{lll}1 & 0 & 0 \\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\).

Step-by-Step Solution

Answer

By following the process of matrix multiplication and addition, we found that the equation \(E^{2} F+F^{2} E=E\) indeed holds true.

Step 1: Find the product of E squared and F

Firstly, to compute \(E^{2} F\), you need to calculate \(E^{2}\) which is the product of matrix E with itself. Once that is done, multiply the result with matrix F. This is done by taking each row of \(E^{2}\) and multiplying it corresponding element of each column in F, and summing them all up to get the elements of the product matrix.

Step 2: Find the product of F squared and E

Secondly, to calculate \(F^{2} E\), first calculate \(F^{2}\) which is the product of matrix F with itself and then multiply the resulting matrix by matrix E. The multiplication is similar to step 1.

Step 3: Add the results from Step 1 and Step 2

After calculating \(E^{2} F\) and \(F^{2} E\), add these two matrices together. This is done by summing the corresponding elements in each matrix.

Step 4: Verify the result

Finally, compare the resulting matrix from step 3 to matrix E. If they are the same, then the equation \(E^{2} F+F^{2} E=E\) holds true, otherwise, it's not valid.

Key Concepts

Matrix Squaring

Matrix squaring is a process of multiplying a matrix by itself. It’s a specific case of matrix multiplication. In our example, we need to square two matrices: matrix \( E \) and matrix \( F \). Let's delve deeper into how this works.
For a matrix square, each element in the resulting matrix \( A^2 \) is computed by taking the dot product of the corresponding row and column in matrix \( A \).

• Each element in the resulting matrix is calculated by multiplying each element of a row by each corresponding element of a column, and then summing the results.
• In our case, the matrix \( E \) is squared first, which means \( E^2 = E \times E \).
• Similarly, for matrix \( F \), squaring it results in \( F^2 = F \times F \).

This operation is foundational in linear algebra and is used to perform transformations and solve systems of equations in various applications.

Matrix Addition

Matrix addition involves taking two matrices of the same dimensions and producing a new matrix by adding their corresponding elements. This operation is fairly straightforward but essential to understand when dealing with complex problems like our exercise.
To properly add matrices such as \( E^2F \) and \( F^2E \), we follow these steps:

• Ensure both matrices are of the same order or dimensions.
• Add the corresponding elements of the matrices together.

In our example:
The result from squaring and multiplying the matrices \( E \) and \( F \) gives us two matrices \( E^2F \) and \( F^2E \). By adding these matrices together, we combine their results element-wise. The addition process must cover every element from the top-left to the bottom-right, thoroughly making sure each entry aligns properly.

Verification of Matrix Equation

The verification process is the step where we check if the mathematical operations and assumptions are correct. Here, we need to verify if \( E^2 F + F^2 E = E \).
This verification process involves:

• Calculating \( E^2F \)
• Calculating \( F^2E \)
• Adding both of these results together to see if they match matrix \( E \).

If after adding \( E^2F \) and \( F^2E \) the result is precisely the original matrix \( E \), then the verification is positive and confirms that the equation holds true.
In linear algebra, verification is crucial for confirming that transformations or operations are functioning as expected and is often used to prove theoretical properties of matrices in geometric transformations or system modeling.