Problem 54 - 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. $$P=\left[\begin{array}{ll}{3} & {4} \\ {1} & {2}\end{array}\right] \quad Q=\left[\begin{array}{rr}{-1} & {0} \\ {3} & {-2}\end{array}\right] \quad R=\left[\begin{array}{rr}{1} & {4} \\ {-2} & {1}\end{array}\right] \quad S=\left[\begin{array}{ll}{0} & {1} \\ {2} & {0}\end{array}\right] \quad I=\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]$$ $$ (P+Q) I \text { and } P I+Q I $$

Step-by-Step Solution

Answer

(P+Q)I and PI+QI are equal because multiplying a matrix by the identity matrix I does not change the original matrix.

Step 1: Understand the Identity Matrix

The identity matrix acts as the multiplicative identity in matrix multiplication. This means that any matrix multiplied by the identity matrix (I) will result in the original matrix. So, for any matrix A, the equation AI = A holds true.

Step 2: Multiply (P+Q) by Identity Matrix (I)

First, we sum matrices P and Q, then we multiply the resulting matrix by I. Adding P and Q gives us a new matrix, and multiplying by the identity matrix will yield the same matrix because AI = A.

Step 3: Multiply P and Q by Identity Matrix (I) Individually

We multiply matrices P and Q by the identity matrix I individually to get PI and QI. Since multiplying by I does not change the matrix, PI will equal P and QI will equal Q.

Step 4: Compare the results

Now we can compare (P+Q)I from Step 2 with PI+QI from Step 3. If the results are equal, then the expressions are equal.

Key Concepts

Identity Matrix

Understanding the identity matrix is crucial when working with matrix algebra. It is the equivalent of the number one in regular multiplication. The identity matrix, often denoted as 'I', is a square matrix in which all the elements of the principal diagonal are ones, and all other elements are zeros. For example, a 2x2 identity matrix looks like this:
\[I = \begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\]
When you multiply any matrix by the identity matrix, the original matrix remains unchanged. This property holds true for matrices of any size, provided that the number of columns in the matrix matches the number of rows in the identity matrix. The formula can be expressed succinctly as AI = IA = A, where A represents any matrix. Think of the identity matrix as a mirror that reflects any matrix back upon itself.

Matrix Addition

Matrix addition is another fundamental concept in the world of matrix algebra. The operation is straightforward: you can add two matrices by adding their corresponding elements. However, this requires that both matrices have the same dimensions. For instance, a 2x2 matrix can only be added to another 2x2 matrix. Here's how you would add two matrices P and Q:
\[\begin{bmatrix}a_1 & b_1 \c_1 & d_1\end{bmatrix} + \begin{bmatrix}a_2 & b_2 \c_2 & d_2\end{bmatrix} = \begin{bmatrix}a_1 + a_2 & b_1 + b_2 \c_1 + c_2 & d_1 + d_2\end{bmatrix}\]
In the context of the exercise provided, adding matrix P to Q would give us a new matrix that highlights the cumulative effect of each element from P and Q. As an exercise improvement, it's essential not to rush through this process. Carefully align the matrices and add each corresponding element to ensure that your resulting matrix is accurate.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the operations that can be performed on them, including addition, multiplication, and scalar multiplication. It's a powerful tool used in various fields such as physics, engineering, computer science, and economy. One of the key features of matrix algebra is that these operations must follow specific rules, very much like the arithmetic for numbers, but with an added layer of complexity due to the nature of matrices.
When dealing with matrix multiplication, remember that the order in which you multiply matrices matters, unlike number multiplication. For matrices A and B, generally, AB is not equal to BA. In the provided exercise, by understanding how these operations work and following the steps, students can confidently solve equations involving matrix algebra. To enhance understanding, always double check each step for accuracy, and remember to consider the dimensions of each matrix when performing operations, as this could determine whether the operation is defined or not.