Problem 34 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of \(A^2\) is \(\begin{bmatrix} 200 & 300 \\ 75 & 725 \end{bmatrix}\).

Step 1: Understand the Operation

To find \(A^2\), we need to multiply matrix \(A\) by itself. Matrix multiplication is defined only if the number of columns in the first matrix equals the number of rows in the second matrix.

Step 2: Confirm Dimensions

Matrix \(A\) is a 2x2 matrix. When multiplying \(A\) by \(A\), both matrices have 2 columns and 2 rows, respectively. Hence, the multiplication is possible, and the resulting product will also be a 2x2 matrix.

Step 3: Multiply the Matrices

Perform the multiplication of matrix \(A\) by itself:\[A \times A = \begin{bmatrix}-10 & 20 \5 & 25\end{bmatrix}\times\begin{bmatrix}-10 & 20 \5 & 25\end{bmatrix}\]Calculate each element of the resulting matrix:- First row, first column: \((-10)\times(-10) + 20\times5 = 100 + 100 = 200\)- First row, second column: \((-10)\times20 + 20\times25 = -200 + 500 = 300\)- Second row, first column: \(5\times(-10) + 25\times5 = -50 + 125 = 75\)- Second row, second column: \(5\times20 + 25\times25 = 100 + 625 = 725\)

Step 4: Write the Resulting Matrix

Based on the calculations, the resulting matrix \(A^2\) is:\[A^2 = \begin{bmatrix}200 & 300 \75 & 725\end{bmatrix}\]

Key Concepts

2x2 matrices

A 2x2 matrix is one of the simplest forms of a matrix structure with two rows and two columns. It's typically written in the form: \(\begin{bmatrix}a & b \c & d\end{bmatrix}\), where \(a, b, c,\) and \(d\) are elements of the matrix. These elements can be any type of numbers, such as integers or real numbers.
2x2 matrices are often used in various mathematical operations, such as addition, subtraction, and multiplication. They are especially useful for representing linear equations and transformations due to their simple structure.

• Each element in a 2x2 matrix represents an entry in a grid.
• 2x2 matrices are square matrices because they have an equal number of rows and columns.
• Common notation includes naming matrices with capital letters like \(A, B,\) or \(C\).

Understanding 2x2 matrices is fundamental when exploring more complex matrix operations.

matrix operations

Matrix operations encompass a variety of mathematical processes that can be applied to matrices, such as addition, subtraction, and multiplication. Each operation has its own set of rules.
Among these, matrix multiplication is one of the more complex procedures that involves combining two matrices to produce a new matrix. The rule for matrix multiplication states that you can only multiply matrices if the number of columns of the first matrix equals the number of rows of the second.

• In multiplication, the order of matrices matters (i.e., \(AB eq BA\)).
• Each element of the resulting matrix is calculated as the dot product of corresponding rows and columns.
• For 2x2 matrices, the calculations involve simple arithmetic, yet careful attention to detail is necessary.

Practicing matrix operations can significantly help in understanding larger systems and transformations in linear algebra.

square matrix

A square matrix is defined as having the same number of rows and columns. This characteristic makes it a special type of matrix.
In the context of 2x2 matrices, they automatically qualify as square matrices. Square matrices have interesting properties, like the possibility of being invertible (if they have a non-zero determinant) or having special calculations like determinants and eigenvalues.

• Square matrices have symmetric dimensions, making certain operations more straightforward.
• They form the basis for more complex mathematical concepts such as determinants.
• In square matrices, an operation like \(A^2\) simply means multiplying the matrix by itself.

Understanding square matrices is crucial, as their unique features often simplify many advanced mathematical processes.