Problem 75
Question
Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right]\), find each product if possible. $$A^{2}$$
Step-by-Step Solution
Verified Answer
\( A^2 = \begin{bmatrix} 10 & -10 \\ 15 & -5 \end{bmatrix} \)
1Step 1: Understand the Problem
To find \( A^2 \), we need to multiply matrix \( A \) by itself. Matrix \( A \) is a \( 2 \times 2 \) matrix, so its square will also be a \( 2 \times 2 \) matrix.
2Step 2: Set Up the Multiplication
Write out the multiplication of \( A \cdot A \), as follows: \ \( A \times A = \left[\begin{array}{rr} 4 & -2 \ 3 & 1 \end{array}\right] \times \left[\begin{array}{rr} 4 & -2 \ 3 & 1 \end{array}\right] \).
3Step 3: Perform the Matrix Multiplication
The element in the first row, first column of the product \( A^2 \) is computed as \( 4 \times 4 + (-2) \times 3 = 16 - 6 = 10 \).The element in the first row, second column is \( 4 \times (-2) + (-2) \times 1 = -8 - 2 = -10 \).The element in the second row, first column is \( 3 \times 4 + 1 \times 3 = 12 + 3 = 15 \).The element in the second row, second column is \( 3 \times (-2) + 1 \times 1 = -6 + 1 = -5 \).
4Step 4: Write the Resultant Matrix
The product \( A^2 \) is \[ A^2 = \begin{bmatrix} 10 & -10 \ 15 & -5 \end{bmatrix} \].
Key Concepts
Matrix ProductsSquare MatrixMatrix Algebra
Matrix Products
Matrix products, often referred to as matrix multiplication, is a fundamental operation in linear algebra. This operation involves combining two matrices to produce another matrix. The process is somewhat like multiplying numbers, but instead we multiply the rows of the first matrix by the columns of the second matrix.
To multiply matrices, it's important that the number of columns in the first matrix matches the number of rows in the second matrix. For instance, if you multiply a matrix with dimensions \(m \times n\) by another matrix with dimensions \(n \times p\), the resulting matrix will have dimensions \(m \times p\).
When it comes to the exercise involving matrix \(A\), since \(A\) is a square matrix with dimensions \(2 \times 2\), multiplying it by itself, \(A^2\), is straightforward. Each element in the resulting matrix \(A^2\) is obtained by summing the products of corresponding elements from the rows of the first \(A\) and the columns of the second \(A\). This results in a new \(2 \times 2\) matrix.
To multiply matrices, it's important that the number of columns in the first matrix matches the number of rows in the second matrix. For instance, if you multiply a matrix with dimensions \(m \times n\) by another matrix with dimensions \(n \times p\), the resulting matrix will have dimensions \(m \times p\).
When it comes to the exercise involving matrix \(A\), since \(A\) is a square matrix with dimensions \(2 \times 2\), multiplying it by itself, \(A^2\), is straightforward. Each element in the resulting matrix \(A^2\) is obtained by summing the products of corresponding elements from the rows of the first \(A\) and the columns of the second \(A\). This results in a new \(2 \times 2\) matrix.
Square Matrix
A square matrix is a matrix with the same number of rows and columns. Examples include matrices with dimensions like \(2 \times 2\), \(3 \times 3\), and so on. Square matrices are significant in matrix algebra because operations like finding the determinant, eigenvalues, and matrix exponentiation are typically reserved for square matrices.
In the context of this problem, matrix \(A\) is a \(2 \times 2\) square matrix. This means we can perform matrix \(A^2\), a special operation called squaring the matrix, which involves multiplying the matrix by itself. The phrase "squaring a matrix" becomes meaningful only when the matrix is square, as the resulting matrix will also be square with the same dimensions.
In the context of this problem, matrix \(A\) is a \(2 \times 2\) square matrix. This means we can perform matrix \(A^2\), a special operation called squaring the matrix, which involves multiplying the matrix by itself. The phrase "squaring a matrix" becomes meaningful only when the matrix is square, as the resulting matrix will also be square with the same dimensions.
- If we were tasked with squaring a non-square matrix, it wouldn't be possible with standard matrix multiplication rules.
- Square matrices sometimes have unique properties, like having an inverse or determinant (if not singular).
Matrix Algebra
Matrix algebra forms the backbone of operations involving matrices, providing rules and formulas for how matrices can be added, subtracted, multiplied, and transformed. It's analogous to regular algebra with numbers but much broader due to the multi-dimensional nature of matrices.
For this exercise, understanding matrix algebra is crucial because it allows us to perform operations like matrix products correctly. In matrix multiplication, the rules of matrix algebra guide how each element in the rows and columns are multiplied and added to form the new matrix.
Some key aspects of matrix algebra include:
For this exercise, understanding matrix algebra is crucial because it allows us to perform operations like matrix products correctly. In matrix multiplication, the rules of matrix algebra guide how each element in the rows and columns are multiplied and added to form the new matrix.
Some key aspects of matrix algebra include:
- Commutative Property: Matrix multiplication is not commutative, meaning \(A \cdot B eq B \cdot A\) in most cases.
- Associative Property: Matrix multiplication is associative: \((A \cdot B) \cdot C = A \cdot (B \cdot C)\).
- Distributive Property: It is distributive: \(A \cdot (B + C) = A \cdot B + A \cdot C\).
Other exercises in this chapter
Problem 75
In your own words, explain what happens when you apply Cramer's rule if \(D=0\)
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A company produces two models of bicycles: model \(A\) and model \(B\). Model \(A\) requires 2 hours of assembly time, and model \(B\) requires 3 hours of assem
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Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=\log (x+5)\\\&y=x^{2}\end{aligned}$$
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Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq \frac{1}{x+2}\\\&y \geq x-2\end{aligned}$$
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