Problem 31
Question
Use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B A\)
Step-by-Step Solution
Verified Answer
The operation is possible; the result is \( \begin{bmatrix} -350 & 1050 \\ 350 & 350 \end{bmatrix} \).
1Step 1: Determine Matrix Compatibility
To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix B is a 2x2 matrix, and matrix A is also a 2x2 matrix. Therefore, matrix multiplication is possible since both matrices have dimensions that support the operation.
2Step 2: Perform Matrix Multiplication
Multiply each element of the rows in matrix B by the corresponding elements of the columns in matrix A, then sum the products to find each element of the resulting matrix.For element (1,1) of BA: \( 40 \times (-10) + 10 \times 5 = -400 + 50 = -350 \)For element (1,2) of BA: \( 40 \times 20 + 10 \times 25 = 800 + 250 = 1050 \)For element (2,1) of BA: \( -20 \times (-10) + 30 \times 5 = 200 + 150 = 350 \)For element (2,2) of BA: \( -20 \times 20 + 30 \times 25 = -400 + 750 = 350 \)
3Step 3: Construct the Resulting Matrix
Based on the calculations, the resulting matrix \( BA \) is:\[BA = \begin{bmatrix} -350 & 1050 \ 350 & 350 \end{bmatrix}\]
Key Concepts
Matrix Compatibility2x2 MatrixMatrix OperationsResulting Matrix
Matrix Compatibility
Before you can multiply two matrices, you must ensure they are compatible for multiplication. Matrix compatibility determines whether one matrix can be multiplied by another.
To check compatibility, look at two critical aspects:
For instance, in the given exercise, both matrices A and B are 2x2 matrices, meaning each has 2 rows and 2 columns. This makes them compatible, so multiplication can proceed.
To check compatibility, look at two critical aspects:
- The number of columns in the first matrix.
- The number of rows in the second matrix.
For instance, in the given exercise, both matrices A and B are 2x2 matrices, meaning each has 2 rows and 2 columns. This makes them compatible, so multiplication can proceed.
2x2 Matrix
A 2x2 matrix consists of 2 rows and 2 columns. Each cell in the matrix contains a specific numeric value. These matrices are simple and commonly used in many calculations due to their manageable size.
In the matrices from the exercise, Matrix A is \[A = \begin{bmatrix}-10 & 20 \5 & 25\end{bmatrix}\] and Matrix B is \[B = \begin{bmatrix}40 & 10 \-20 & 30\end{bmatrix}\]. They both have 2 rows and 2 columns.
This makes performing operations like addition, subtraction, and multiplication very convenient.
In the matrices from the exercise, Matrix A is \[A = \begin{bmatrix}-10 & 20 \5 & 25\end{bmatrix}\] and Matrix B is \[B = \begin{bmatrix}40 & 10 \-20 & 30\end{bmatrix}\]. They both have 2 rows and 2 columns.
This makes performing operations like addition, subtraction, and multiplication very convenient.
Matrix Operations
Matrix operations such as addition, subtraction, and especially multiplication are fundamental in linear algebra.
For multiplication, specifically, the value of each element in the resulting matrix is found through the dot product of the corresponding row of the first matrix and the corresponding column of the second matrix.
To compute the matrix product, such as the product of matrix B and matrix A:
This step is repeated for all elements of the resulting matrix.
For multiplication, specifically, the value of each element in the resulting matrix is found through the dot product of the corresponding row of the first matrix and the corresponding column of the second matrix.
To compute the matrix product, such as the product of matrix B and matrix A:
- Multiply the elements of the rows of B by the corresponding elements of the columns of A.
- Sum these products to get the elements of the new matrix.
This step is repeated for all elements of the resulting matrix.
Resulting Matrix
After performing matrix multiplication, you'll obtain a new matrix called the resulting matrix. This matrix has its dimensions determined by the number of rows of the first matrix and the number of columns of the second matrix.
Given matrices A and B, the resulting matrix after their multiplication, denoted as BA, is also a 2x2 matrix:\[BA = \begin{bmatrix}-350 & 1050 \350 & 350\end{bmatrix}\]
Each element within this resulting matrix was calculated using the respective elements from the original matrices B and A, following the rules of matrix multiplication.
This illustrates how compatible matrices can transform into a new matrix with potentially new properties and interpretations.
Given matrices A and B, the resulting matrix after their multiplication, denoted as BA, is also a 2x2 matrix:\[BA = \begin{bmatrix}-350 & 1050 \350 & 350\end{bmatrix}\]
Each element within this resulting matrix was calculated using the respective elements from the original matrices B and A, following the rules of matrix multiplication.
This illustrates how compatible matrices can transform into a new matrix with potentially new properties and interpretations.
Other exercises in this chapter
Problem 31
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