Calculating P + Q Squared Plus 3I Matrix Operations Explained

In the realm of linear algebra, matrix operations are fundamental tools for solving a wide array of problems in mathematics, physics, engineering, and computer science. This article delves into the intricate process of calculating $(P + Q)^2 + 3I$, where $P$ and $Q$ are given matrices, and $I$ is the identity matrix. We will meticulously break down each step, providing a comprehensive understanding of matrix addition, matrix multiplication, and scalar multiplication. This exploration will not only enhance your computational skills but also solidify your grasp of the underlying concepts of matrix algebra.

1. Defining the Matrices

Before we embark on the calculations, let's clearly define the matrices $P$ and $Q$ as provided:

$P = \begin{bmatrix} 2 & 1 & -2 \\ 3 & -1 & 1 \\ 2 & 4 & 1 \end{bmatrix}$

$Q = \begin{bmatrix} 3 & -2 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & -2 \end{bmatrix}$

These matrices, $P$ and $Q$, are both 3x3 matrices, meaning they have three rows and three columns. This dimensionality is crucial as it dictates the rules for matrix addition and multiplication. For instance, two matrices can only be added if they have the same dimensions. Similarly, for matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.

The identity matrix, denoted by $I$, plays a special role in matrix algebra, analogous to the number 1 in scalar algebra. It is a square matrix with 1s on the main diagonal and 0s elsewhere. The size of the identity matrix must be compatible with the matrix it is being multiplied with. In this case, since $P$ and $Q$ are 3x3 matrices, we will use the 3x3 identity matrix:

$I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

The identity matrix has the property that when multiplied by any matrix of compatible dimensions, it leaves the matrix unchanged. That is, for any matrix $A$, $AI = IA = A$.

2. Matrix Addition: P + Q

The first step in our calculation is to add the matrices $P$ and $Q$. Matrix addition is a straightforward operation that involves adding corresponding elements of the matrices. In other words, we add the element in the $i$-th row and $j$-th column of $P$ to the element in the $i$-th row and $j$-th column of $Q$. This can be expressed as:

$(P + Q)_{ij} = P_{ij} + Q_{ij}$

Applying this rule to our matrices, we get:

$P + Q = \begin{bmatrix} 2 & 1 & -2 \\ 3 & -1 & 1 \\ 2 & 4 & 1 \end{bmatrix} + \begin{bmatrix} 3 & -2 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & -2 \end{bmatrix} = \begin{bmatrix} 2+3 & 1+(-2) & -2+1 \\ 3+2 & -1+1 & 1+1 \\ 2+1 & 4+1 & 1+(-2) \end{bmatrix}$

Simplifying the element-wise additions, we obtain:

$P + Q = \begin{bmatrix} 5 & -1 & -1 \\ 5 & 0 & 2 \\ 3 & 5 & -1 \end{bmatrix}$

This resulting matrix, $(P + Q)$, is the sum of the matrices $P$ and $Q$. It is also a 3x3 matrix, as expected, since we added two 3x3 matrices. The elements of $(P + Q)$ represent the combined effect of the corresponding elements in $P$ and $Q$.

3. Squaring the Sum: (P + Q)^2

Now that we have calculated $(P + Q)$, the next step is to square this matrix, which means multiplying it by itself. Matrix multiplication is a more complex operation than matrix addition. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. If $A$ is an $m imes n$ matrix and $B$ is an $n imes p$ matrix, then their product, $AB$, is an $m imes p$ matrix. The element in the $i$-th row and $j$-th column of $AB$ is obtained by taking the dot product of the $i$-th row of $A$ and the $j$-th column of $B$.

In our case, we need to calculate $(P + Q)^2 = (P + Q)(P + Q)$. Since $(P + Q)$ is a 3x3 matrix, we can multiply it by itself. Let's denote $(P + Q)$ as $R$ for simplicity:

$R = P + Q = \begin{bmatrix} 5 & -1 & -1 \\ 5 & 0 & 2 \\ 3 & 5 & -1 \end{bmatrix}$

Then, we need to calculate $R^2 = R imes R$:

$R^2 = \begin{bmatrix} 5 & -1 & -1 \\ 5 & 0 & 2 \\ 3 & 5 & -1 \end{bmatrix} \begin{bmatrix} 5 & -1 & -1 \\ 5 & 0 & 2 \\ 3 & 5 & -1 \end{bmatrix}$

To compute the elements of $R^2$, we perform the dot product of the rows of the first matrix with the columns of the second matrix. For example, the element in the first row and first column of $R^2$ is:

$(R^2)_{11} = (5)(5) + (-1)(5) + (-1)(3) = 25 - 5 - 3 = 17$

Similarly, we can calculate the other elements:

$(R^2)_{12} = (5)(-1) + (-1)(0) + (-1)(5) = -5 - 0 - 5 = -10$

$(R^2)_{13} = (5)(-1) + (-1)(2) + (-1)(-1) = -5 - 2 + 1 = -6$

$(R^2)_{21} = (5)(5) + (0)(5) + (2)(3) = 25 + 0 + 6 = 31$

$(R^2)_{22} = (5)(-1) + (0)(0) + (2)(5) = -5 + 0 + 10 = 5$

$(R^2)_{23} = (5)(-1) + (0)(2) + (2)(-1) = -5 + 0 - 2 = -7$

$(R^2)_{31} = (3)(5) + (5)(5) + (-1)(3) = 15 + 25 - 3 = 37$

$(R^2)_{32} = (3)(-1) + (5)(0) + (-1)(5) = -3 + 0 - 5 = -8$

$(R^2)_{33} = (3)(-1) + (5)(2) + (-1)(-1) = -3 + 10 + 1 = 8$

Therefore, the resulting matrix $R^2$ is:

$(P + Q)^2 = R^2 = \begin{bmatrix} 17 & -10 & -6 \\ 31 & 5 & -7 \\ 37 & -8 & 8 \end{bmatrix}$

This matrix represents the square of the sum of matrices $P$ and $Q$. Each element in this matrix is a result of the combined interactions of the elements in $(P + Q)$ through the process of matrix multiplication.

4. Scalar Multiplication: 3I

The next step is to calculate $3I$, where $I$ is the 3x3 identity matrix. Scalar multiplication involves multiplying a matrix by a scalar (a single number). To do this, we simply multiply each element of the matrix by the scalar. In our case, we need to multiply the identity matrix $I$ by the scalar 3:

$3I = 3 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Multiplying each element of $I$ by 3, we get:

$3I = \begin{bmatrix} 3(1) & 3(0) & 3(0) \\ 3(0) & 3(1) & 3(0) \\ 3(0) & 3(0) & 3(1) \end{bmatrix} = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

This resulting matrix, $3I$, is a diagonal matrix with 3s on the main diagonal and 0s elsewhere. Scalar multiplication scales the matrix by the given scalar, effectively multiplying each element by that factor. In this case, multiplying by 3 has tripled the values on the main diagonal of the identity matrix.

5. Final Calculation: (P + Q)^2 + 3I

Finally, we can calculate $(P + Q)^2 + 3I$ by adding the matrix $(P + Q)^2$ that we calculated earlier to the matrix $3I$. As with the addition of $P$ and $Q$, adding matrices $(P + Q)^2$ and $3I$ involves adding corresponding elements of the matrices:

$(P + Q)^2 + 3I = \begin{bmatrix} 17 & -10 & -6 \\ 31 & 5 & -7 \\ 37 & -8 & 8 \end{bmatrix} + \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

Adding the corresponding elements, we get:

$(P + Q)^2 + 3I = \begin{bmatrix} 17+3 & -10+0 & -6+0 \\ 31+0 & 5+3 & -7+0 \\ 37+0 & -8+0 & 8+3 \end{bmatrix}$

Simplifying the element-wise additions, we obtain:

$(P + Q)^2 + 3I = \begin{bmatrix} 20 & -10 & -6 \\ 31 & 8 & -7 \\ 37 & -8 & 11 \end{bmatrix}$

This is the final result of our calculation. The matrix $(P + Q)^2 + 3I$ represents the sum of the square of the sum of matrices $P$ and $Q$ and three times the identity matrix. This result combines the operations of matrix addition, matrix multiplication, and scalar multiplication, demonstrating the power and versatility of matrix algebra.

Conclusion

In this comprehensive guide, we have meticulously walked through the process of calculating $(P + Q)^2 + 3I$, where $P$ and $Q$ are given matrices and $I$ is the identity matrix. We have explored the fundamental operations of matrix addition, matrix multiplication, and scalar multiplication. By breaking down each step and providing detailed explanations, we have aimed to enhance your understanding of matrix algebra and your ability to perform these calculations with confidence. Mastering these matrix operations is crucial for tackling more advanced topics in linear algebra and its applications in various fields.