Problem 56 - Algebra 2

Question

For Exercises \(53-56,\) use matrices \(P, Q, R, S,\) and \(I .\) Determine whether the two expressions in each pair are equal. $$P=\left[\begin{array}{ll}{3} & {4} \\ {1} & {2}\end{array}\right] \quad Q=\left[\begin{array}{rr}{-1} & {0} \\ {3} & {-2}\end{array}\right] \quad R=\left[\begin{array}{rr}{1} & {4} \\ {-2} & {1}\end{array}\right] \quad S=\left[\begin{array}{ll}{0} & {1} \\ {2} & {0}\end{array}\right] \quad I=\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]$$ $$ (P+Q)(R+S) \text { and } P R+P S+Q R+Q S $$

Step-by-Step Solution

Answer

After performing the calculations, the expressions \(P+Q)(R+S)\ and \(PR + PS + QR + QS)\ are equal if their resulting matrices are identical.

Step 1: Matrix Addition (P + Q)

Compute the sum of matrices P and Q by adding the corresponding elements of each matrix.

Step 2: Matrix Addition (R + S)

Find the sum of matrices R and S by adding their corresponding elements.

Step 3: Matrix Multiplication ((P + Q)(R + S))

Multiply the resulting matrices (P + Q) and (R + S) using the rules of matrix multiplication.

Step 4: Matrix Multiplications (PR, PS, QR, QS)

Compute each of the matrix products PR, PS, QR, and QS individually.

Step 5: Matrix Addition (PR + PS + QR + QS)

Add the products from Step 4 together.

Step 6: Compare Results

Compare the resulting matrices from Steps 3 and 5 to determine if the expressions are equal.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in algebra that involves combining two matrices by adding their corresponding elements. To perform this operation, both matrices must be of the same dimension, meaning they must have the same number of rows and columns. Let's take matrices P and Q from our exercise as an example.

For matrix addition, we add each element from the same position in both matrices together. In our case, to calculate P + Q, we compute it like so:

• P[1,1] + Q[1,1] = 3 + (-1) = 2
• P[1,2] + Q[1,2] = 4 + 0 = 4
• P[2,1] + Q[2,1] = 1 + 3 = 4
• P[2,2] + Q[2,2] = 2 + (-2) = 0

The result of P + Q will be a new matrix where each element is the sum of corresponding elements of P and Q. Similarly, we find the sum of matrices R and S. Now, let's breakdown what happens when we're faced with matrix algebra expressions. The expression (P + Q)(R + S) first requires us to find P + Q and R + S as we just described, and only then can we go ahead with the multiplication.

Matrix Multiplication

Matrix multiplication is another core concept of matrix operations that differs significantly from matrix addition. Here, the elements are not simply added together; instead, multiplication combines rows and columns in a more complex manner. The number of columns in the first matrix must match the number of rows in the second matrix for the multiplication to be possible.

In our example, to compute the product of two matrices, say (P + Q) and (R + S), we follow these steps:

1. Multiply each element of the rows of (P + Q) by the corresponding elements of the columns of (R + S).
2. Sum the results of these multiplications to get a single number, which becomes one element in the resulting matrix.
3. Repeat this process for each element in the resultant matrix's rows and columns.

This process yields a new matrix where each element is a product of a row from the first matrix and a column from the second matrix. To understand the concept thoroughly, it's essential to work through an example like the one provided in the exercise, as it helps grasp the multi-step process involved in matrix multiplication.

Matrix Equality

Understanding matrix equality is essential when comparing the results of matrix operations like addition and multiplication. Two matrices are equal if and only if they have the same dimensions and their corresponding elements are equal. It is crucial to ensure that after performing operations on matrices, like those in our textbook exercise, the resulting matrices are checked for equality element by element.

In our case, after calculating the product (P + Q)(R + S) and the sum of products PR + PS + QR + QS, we must compare each corresponding element. If all corresponding elements are indeed equal, then we can confidently say the two matrix expressions result in equal matrices. This step is vital as it verifies whether distributive properties or other algebraic manipulations hold true for the given matrices.Whenever you're faced with complex matrix expressions, remember to break them down part by part, ensuring you follow the rules for equality strictly. Through careful comparison and verification, we either confirm or deny the equivalence of the matrix expressions.