Given The Matrix $A = \begin{pmatrix} 1 & 0 & 1 \\ 2 & -1 & 1 \\ -1 & 1 & -1 \end{pmatrix}$, (a) Calculate The Matrices $A^2$ And $A^3$. (b) Verify That $A^3 + A^2 - A = I_3$. Show The Calculation Steps Clearly.

Matrix calculations can seem daunting at first, but with a systematic approach, you can easily tackle them. In this article, we'll delve into the properties of a specific matrix, A, through a step-by-step calculation of its powers and a verification of a crucial matrix equation. We'll break down each step, making it easy to follow and understand, even if you're new to matrix algebra. So, let's embark on this mathematical journey and unlock the secrets of matrix A.

1. Matrix Definition

First, let's define the matrix A that we'll be working with:

A = \begin{pmatrix}
1 & 0 & 1 \\
2 & -1 & 1 \\
-1 & 1 & -1
\end{pmatrix}

This 3x3 matrix forms the foundation of our exploration. We'll be performing various operations on it, including calculating its square (A²) and cube (A³), and then verifying a specific matrix equation. Understanding the structure of this matrix is key to grasping the subsequent calculations. Each element plays a crucial role in the final results, so paying close attention to the arrangement and values is essential.

2. Calculating A²

To calculate $A^2$, we need to multiply the matrix A by itself. Matrix multiplication involves taking the dot product of the rows of the first matrix with the columns of the second matrix. Let's break this down step-by-step:

$A^2 = A * A = \begin{pmatrix} 1 & 0 & 1 \\ 2 & -1 & 1 \\ -1 & 1 & -1 \end{pmatrix} * \begin{pmatrix} 1 & 0 & 1 \\ 2 & -1 & 1 \\ -1 & 1 & -1 \end{pmatrix}$

To find the element in the first row and first column of $A^2$, we calculate:

(1 * 1) + (0 * 2) + (1 * -1) = 1 + 0 - 1 = 0

Similarly, for the element in the first row and second column:

(1 * 0) + (0 * -1) + (1 * 1) = 0 + 0 + 1 = 1

Continuing this process for all elements, we get:

$A^2 = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 2 & 0 \\ 2 & -2 & 1 \end{pmatrix}$

The calculation of $A^2$ is a fundamental step. It showcases the process of matrix multiplication, which is crucial for many linear algebra operations. Each element in the resulting matrix is a product of a row and a column from the original matrix, making this operation a cornerstone of matrix algebra. By carefully applying the rules of matrix multiplication, we've successfully computed $A^2$, paving the way for the next step in our exploration.

3. Calculating A³

Now that we have $A^2$, we can calculate $A^3$ by multiplying $A^2$ by A:

$A^3 = A^2 * A = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 2 & 0 \\ 2 & -2 & 1 \end{pmatrix} * \begin{pmatrix} 1 & 0 & 1 \\ 2 & -1 & 1 \\ -1 & 1 & -1 \end{pmatrix}$

Again, we apply the rules of matrix multiplication. For the element in the first row and first column of $A^3$:

(0 * 1) + (1 * 2) + (0 * -1) = 0 + 2 + 0 = 2

For the element in the first row and second column:

(0 * 0) + (1 * -1) + (0 * 1) = 0 - 1 + 0 = -1

Continuing for all elements:

$A^3 = \begin{pmatrix} 2 & -1 & 1 \\ 3 & -2 & 1 \\ -1 & 1 & 0 \end{pmatrix}$

Calculating $A^3$ builds upon our understanding of matrix multiplication. It demonstrates how we can extend the concept of squaring a matrix to higher powers. This step is crucial as it provides us with the necessary components to verify the given matrix equation. The process of multiplying $A^2$ by A reinforces the mechanics of matrix multiplication, ensuring a solid grasp of the operation. The resulting matrix, $A^3$, is a key ingredient in the final verification step, highlighting the importance of accurate calculations.

4. Verifying the Equation A³ + A² - A = I₃

The final step is to verify that $A^3 + A^2 - A = I_3$, where $I_3$ is the 3x3 identity matrix:

$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Let's substitute the calculated matrices into the equation:

$\begin{pmatrix} 2 & -1 & 1 \\ 3 & -2 & 1 \\ -1 & 1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 1 & 0 \\ -1 & 2 & 0 \\ 2 & -2 & 1 \end{pmatrix} - \begin{pmatrix} 1 & 0 & 1 \\ 2 & -1 & 1 \\ -1 & 1 & -1 \end{pmatrix}$

Now, we perform matrix addition and subtraction. Remember, these operations involve adding or subtracting corresponding elements in the matrices:

$\begin{pmatrix} 2+0-1 & -1+1-0 & 1+0-1 \\ 3-1-2 & -2+2-(-1) & 1+0-1 \\ -1+2-(-1) & 1-2-1 & 0+1-(-1) \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & -2 & 1 \end{pmatrix}$

Simplifying, we get:

$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

This is indeed the identity matrix $I_3$. Therefore, we have verified that $A^3 + A^2 - A = I_3$.

Verifying the equation $A^3 + A^2 - A = I_3$ is the culmination of our calculations. This step demonstrates the relationship between different powers of matrix A and the identity matrix. Matrix addition and subtraction, performed element-wise, are crucial operations in this verification. The result confirms a specific property of matrix A, showcasing its unique characteristics. This verification not only validates our calculations but also highlights the importance of understanding matrix algebra in uncovering such relationships.

5. Conclusion

Through this detailed exploration, we've successfully calculated $A^2$ and $A^3$, and verified the equation $A^3 + A^2 - A = I_3$. This exercise provides a solid understanding of matrix operations and their applications. By carefully performing each step, from matrix multiplication to addition and subtraction, we've uncovered a fundamental property of matrix A. This knowledge equips you to tackle more complex matrix problems and appreciate the elegance of linear algebra. Matrix algebra, while sometimes appearing complex, is a powerful tool in various fields, from computer graphics to data analysis. Mastering these fundamental operations is key to unlocking its potential.