Problem 40 - College Algebra

Question

Use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \((B A)^{2}\)

Step-by-Step Solution

Answer

The operation \((BA)^2\) is possible and the result is \(\left[\begin{array}{rr}641250 & 0 \\ 0 & 516250\end{array}\right]\)."

Step 1: Determine if BA is Defined

The matrix multiplication \(BA\) requires that the number of columns in \(B\) is equal to the number of rows in \(A\). Both \(B\) and \(A\) are \(2 \times 2\) matrices, so the multiplication is possible.

Step 2: Multiply Matrices B and A

Multiply the matrices \(B\) and \(A\): \[BA = \left[\begin{array}{rr}40 & 10 \ -20 & 30\end{array}\right] \times \left[\begin{array}{rr}-10 & 20 \ 5 & 25\end{array}\right] = \left[\begin{array}{cc}(40)(-10) + (10)(5) & (40)(20) + (10)(25) \ (-20)(-10) + (30)(5) & (-20)(20) + (30)(25)\end{array}\right] = \left[\begin{array}{rr}-350 & 1125 \ 350 & 350\end{array}\right]\]

Step 3: Calculate \((BA)^2\)

The operation \((BA)^2\) implies the multiplication of \(BA\) by itself: \[(BA)^2 = \left[\begin{array}{rr}-350 & 1125 \ 350 & 350\end{array}\right] \times \left[\begin{array}{rr}-350 & 1125 \ 350 & 350\end{array}\right] = \left[\begin{array}{cc}(-350)(-350) + (1125)(350) & (-350)(1125) + (1125)(350) \ (350)(-350) + (350)(350) & (350)(1125) + (350)(350)\end{array}\right] = \left[\begin{array}{rr}641250 & 0 \ 0 & 516250\end{array}\right]\]

Step 4: Confirm Result

Review each multiplication and addition operation from the previous step to confirm that the calculations are accurate. This checks that the matrix multiplication was carried out correctly.

Key Concepts

2x2 matrix

A 2x2 matrix is a fundamental concept in linear algebra that is represented by a grid containing two rows and two columns. It looks like this:

• Simple Structure: Each entry or element is a number.
• Placement: The first number is positioned in the "first row, first column," the second number in the "first row, second column," and so on.

It can be mathematically written as follows:\[M = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]
Here, each element such as \(a\), \(b\), \(c\), and \(d\) are real numbers. This form allows it to easily be used in mathematical operations like addition, subtraction, and multiplication. Understanding how these small matrices are laid out helps to grasp more complex operations like matrix multiplication.

matrix operations

Matrix operations are ways to manipulate matrices to achieve desired outcomes, much like regular arithmetic with numbers but with specific rules. Some common matrix operations include:

• Addition and Subtraction: This is only possible if the matrices have the same dimensions. Each element in one matrix is added to or subtracted from the corresponding element in the other matrix.
• Scalar Multiplication: Each element of the matrix is multiplied by a single number called the scalar.
• Matrix Multiplication: A more complex operation that requires specific conditions and arrangements.

These operations are essential for simplifying and solving larger mathematical problems, such as those encountered in linear algebra systems.

matrix multiplication definition

Matrix multiplication is a way of combining two matrices to form a new matrix. Unlike scalar multiplication, it is not performed element-wise. Here's how it is done:

• Size Requirement: If matrix \(A\) is of size \(m \times n\), and matrix \(B\) is of size \(n \times p\), matrix multiplication can occur resulting in a new matrix of size \(m \times p\).
• Multiplication Process: Each element of the resulting matrix is computed by taking the dot product of the rows of the first matrix with the columns of the second matrix.

The specific layout of multiplications and additions makes matrix multiplication unique. For example, with the matrices \(A\) and \(B\) mentioned previously:\[AB = \begin{bmatrix} (a_{11})(b_{11}) + (a_{12})(b_{21}) & (a_{11})(b_{12}) + (a_{12})(b_{22}) \ (a_{21})(b_{11}) + (a_{22})(b_{21}) & (a_{21})(b_{12}) + (a_{22})(b_{22}) \end{bmatrix}\]
Remember, order matters in matrix multiplication, \(AB eq BA\). This operation underpins many applications such as computer graphics, engineering, and solving equations.