Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Use a graphing utility to perform the indicated operations. \(A=\left[\begin{array}{ll}2 & 0 \\ 4 & 5\end{array}\right] \quad\) and \(\quad B=\left[\begin{array}{ll}5 & 3 \\ 1 & 4\end{array}\right]\) $$B^{2}-7 B+6 I_{2}$$

Step-by-Step Solution

Answer

The sum after the series of operations on the matrices results in a new 2x2 matrix that answers the matrix operation. The detailed results will depend on the computations done on the graphing utility.

Step 1: Computing Square of Matrix B

To get \(B^{2}\), we multiply matrix B by itself. For any matrix \[A=\left[\begin{array}{ll}a & b \ c & d\end{array}\right] \] , the square is given by \[A^{2}=\left[\begin{array}{ll}a & b \ c & d\end{array}\right] \times \left[\begin{array}{ll}a & b \ c & d\end{array}\right] = \left[\begin{array}{ll} a^{2}+bc & ab+bd \ ac+cd & bc+d^{2}\end{array}\right]\] . Therefore, by substituting values from matrix B, we compute \( B^{2}\).

Step 2: Subtract 7B from \(B^{2}\)

Perform multiplication of Matrix B with the scalar value 7 (\(7B\)), then subtract the result from \(B^{2}\). Remember when you perform scalar multiplication with a matrix, each element inside the matrix is multiplied by the scalar value.

Step 3: Add 6 times the Identity Matrix to the Result

Multiply the 2x2 identity matrix, \(I_{2}\), by 6 and then add it to the matrix obtained from step 2. The 2x2 identity matrix is \[ I_{2}= \left[\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right]\].When a scalar multiple of an identity matrix is added to a matrix, each element on the diagonal of that matrix is added with the scalar. Result from step 2 would be added with \[6 \times I_{2}\] to get the final answer.

Key Concepts

Matrix Multiplication

When we talk about matrix multiplication, we refer to a process where two matrices are combined to produce a third matrix. Unlike scalar multiplication, where we simply multiply each element of a matrix by a constant, matrix multiplication involves a set of rules.

Consider two matrices, A and B. The entry in the i^th row and j^th column of the product matrix AB is calculated by multiplying the elements of the i^th row of A by the corresponding elements of the j^th column of B and then adding up the results. It is crucial that the number of columns in A equals the number of rows in B for the multiplication to be defined.

Here is how matrix multiplication works:

• Match up the rows of the first matrix with the columns of the second matrix.
• Multiply each element of the row by the corresponding element of the column.
• Add these products together to get the single number that goes in the corresponding spot in the resulting matrix.

Each element of the resulting matrix is a sum of products which is calculated independently of the other elements.

Scalar Multiplication

Scalar multiplication involves multiplying every entry in a matrix by a single number, known as a scalar. This operation is straightforward:

For example, if we have a matrix A, and a scalar k, the scalar multiplication of A by k, denoted as kA, will be a matrix where each entry is k times the corresponding entry in A.

To compute it, simply perform the following steps for each element in the matrix:

• Take one element from the matrix.
• Multiply that element by the scalar.
• Place the result in the corresponding position in the new matrix.

The process will result in a new matrix of the same size, with each element reflecting the original value scaled by k.

Identity Matrix

The identity matrix, often denoted as I, plays a similar role in matrix operations as the number 1 does in scalar arithmetic. For a given matrix A, when you multiply it by the identity matrix, you get the same matrix A back.

The identity matrix is a special square matrix that has 1s along the diagonal from the top-left to the bottom-right, and 0s everywhere else, symbolized as:
\[I_{n} = \begin{bmatrix} 1 & 0 & \dots & 0 \ 0 & 1 & \dots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \dots & 1 \end{bmatrix}\]
In matrix operations, the identity matrix is a neutral element. When you add a scalar multiple of an identity matrix to another matrix, the effect is to add that scalar to each diagonal element of the original matrix, leaving other elements unchanged.

Matrix Addition and Subtraction

Matrix addition and subtraction are operations performed element-wise on two matrices of the same dimensions. To add or subtract matrices, simply add or subtract the corresponding elements:

Here's how you do it:

• Take two matrices, A and B, of the same size.
• Add or subtract the elements of A and B that are in the same position.
• Write the result of each addition or subtraction in the corresponding position of the resultant matrix.

It's important to note that we can only add or subtract matrices if they have the same dimensions. If they don't, the operations are not defined. Matrix addition is commutative and associative, similar to how we add numbers, meaning that A + B = B + A and (A + B) + C = A + (B + C). Subtraction, however, is not commutative, so A - B ≠ B - A.