Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Find, if possible, (a) \(A B,\) (b) \(B A,\) and (c) \(A^{2}\). (Note: \(A^{2}=A A\).) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{ll} 1 & 2 \\ 4 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ -1 & 8 \end{array}\right]$$

Step-by-Step Solution

Answer

The results of the matrix operations are: (a) \(AB = \left[\begin{array}{ll} 15 & 0 \ 6 & 12 \end{array}\right]\), (b) \(BA = \left[\begin{array}{ll} -2 & 2 \ 30 & 12 \end{array}\right]\), and (c) \(A^2 = \left[\begin{array}{ll} 9 & 6 \ 12 & 12 \end{array}\right]\)

Step 1: Calculate Matrix A multiplied by Matrix B

The product of matrices A and B is calculated by pairing elements from each row in A with elements from each column in B and adding together. This results in a new matrix: \(AB = \left[\begin{array}{ll} 1(-1)+2*8 & 1*2+2*(-1) \ 4*2+2*(-1) & 4*(-1)+2*8 \end{array}\right] = \left[\begin{array}{ll} 15 & 0 \ 6 & 12 \end{array}\right]\)

Step 2: Calculate Matrix B multiplied by Matrix A

We do this same operation, but swap the order of multiplication from AB to BA. The result will not always be the same as matrix multiplication is not commutative. \(BA = \left[\begin{array}{ll} 2*1+(-1)*4 & 2*2+(-1)*2 \-1*1+8*4 & -1*2 + 8*2\end{array}\right] = \left[\begin{array}{ll} -2 & 2 \ 30 & 12 \end{array}\right]\)

Step 3: Calculate the Square of Matrix A

Here we need to multiply matrix A with itself. \(A^2 = \left[\begin{array}{ll} 1*1+2*4 & 1*2+2*2 \4*1+2*4 & 4*2 +2*2\end{array}\right] = \left[\begin{array}{ll} 9 & 6 \ 12 & 12 \end{array}\right]\)

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. Unlike simple arithmetic multiplication, when multiplying two matrices, we align rows from the first matrix with columns from the second. Each element of the resulting matrix is the product of an entire row and column pair.
For example, to multiply Matrix \(A\) by Matrix \(B\), we take each element from the first row of \(A\) and match it up with each column in \(B\). Now, multiply corresponding elements and add them to get each position's result:

• The first element of the resulting matrix is obtained by multiplying the first row of \(A\) with the first column of \(B\).
• This process is repeated for each row and column intersection, eventually forming a completed matrix.

Hence, matrix multiplication provides us with a unique result that is not symmetric or commutative, meaning the sequence in which you multiply the matrices matters.

Matrix Commutativity

One unique aspect of matrix multiplication is that it is not commutative. In simple terms, commutativity is a property where changing the order of the operation does not change the result. For matrix multiplication, \(AB eq BA\) in most cases.

• In our example, when multiplying Matrix \(A\) by Matrix \(B\), the result was a distinct matrix different from when \(B\) is multiplied by \(A\).
• This order dependency means you should carefully follow the sequence required in your calculations, as reversing it will yield different results.

This principle holds considerable mathematical significance, especially in fields requiring precise calculations, such as computer graphics and complex number systems.

Square of a Matrix

The concept of squaring a matrix is simple but important in linear algebra. Squaring a matrix is akin to raising a number to the power of two, but instead, we multiply the matrix by itself.

• To compute the square of Matrix \(A\), denoted as \(A^2\), multiply Matrix \(A\) by itself.
• Follow the standard matrix multiplication process where each element of the same row and column is multiplied and summed to form the new matrix.

The square of a matrix can provide insight into its properties over repeated processes and is often used in determining powers of matrices for deeper analysis.

Graphing Utility Verification

In the modern technological context, verifying matrix calculations is made easier with graphing utilities. These graphing tools can automatically compute complex matrix operations, reducing human error and saving time.

• These utilities allow you to input the matrix data directly, execute operations like multiplication or squaring, and display the results precisely.
• They act as a learning check, ensuring your manual calculations align with computed results, thus enhancing understanding.

Utilizing graphing utilities is encouraged as it combines learning and practical verification, allowing students to visualize and correct their calculations instantaneously.