Problem 47
Question
Matrix Multiplication Use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$A=\left[\begin{array}{rrr} 1 & -12 & 4 \\ 14 & 10 & 12 \\ 6 & -15 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 12 & 10 \\ -6 & 12 \\ 10 & 16 \end{array}\right]$$
Step-by-Step Solution
Verified Answer
The product of the given matrices, \(A B\), is: \(\left[\begin{array}{cc} 144 & 38 \ 268 & 492 \ 216 & 222 \end{array}\right]\)
1Step 1 - Check for matrix compatibility
Confirm if the two matrices can be multiplied by checking the number of columns in the first matrix, \(A\), and the number of rows in the second matrix, \(B\). For \(A\), we have 3 columns and for \(B\), we have 3 rows. This means the matrices are compatible for multiplication.
2Step 2 - Multiply the matrices
To multiply the matrices, take the dot product of the rows of \(A\) and the columns of \(B\). For example, the entry on the first row and first column of the product matrix will be obtained by multiplying corresponding entries of the first row of \(A\) and first column of \(B\) and adding up the products. Therefore, \(A B = \left[\begin{array}{cc} 1*12+(-12)*(-6)+4*10 & 1*10+(-12)*12+4*16 \ 14*12+10*(-6)+12*10 & 14*10+10*12+12*16 \ 6*12+(-15)*(-6)+3*10 & 6*10+(-15)*12+3*16 \end{array}\right]\)
3Step 3 - Simplify the matrix
Perform the resulting operations in the matrix to simplify it to: \(A B = \left[\begin{array}{cc} 144 & 38 \ 268 & 492 \ 216 & 222 \end{array}\right]\)
Key Concepts
Matrix CompatibilityDot ProductGraphing UtilityMatrix Operations
Matrix Compatibility
Understanding matrix compatibility is crucial when attempting to perform matrix multiplication. It's the first gate through which we must pass to ensure that the operation is possible.
Matrix multiplication isn't like basic arithmetic; we can't just multiply any two matrices together. The rule of thumb for compatibility is that the number of columns in the first matrix must match the number of rows in the second matrix. This condition arises from how we perform the multiplication process, which involves taking the dot product of rows and columns - more on that later.
In our example, matrix A is a 3x3 matrix and matrix B is a 3x2 matrix. Since the number of columns in A matches the number of rows in B, we confirm that A and B are indeed compatible for multiplication. The result will be a new matrix with the number of rows of the first matrix and the number of columns of the second matrix, which in this case, will yield a 3x2 matrix.
Matrix multiplication isn't like basic arithmetic; we can't just multiply any two matrices together. The rule of thumb for compatibility is that the number of columns in the first matrix must match the number of rows in the second matrix. This condition arises from how we perform the multiplication process, which involves taking the dot product of rows and columns - more on that later.
In our example, matrix A is a 3x3 matrix and matrix B is a 3x2 matrix. Since the number of columns in A matches the number of rows in B, we confirm that A and B are indeed compatible for multiplication. The result will be a new matrix with the number of rows of the first matrix and the number of columns of the second matrix, which in this case, will yield a 3x2 matrix.
Dot Product
The dot product is the bread and butter of matrix multiplication. It's a way to multiply corresponding elements from the rows of the first matrix with the columns of the second matrix, and then sum the results to get a single number.
In essence, you multiply each element of a row in the first matrix with each corresponding element of a column in the second matrix and add up all those products. This is done for each element of the resulting matrix.
For example, if we're calculating the entry in the first row and first column of the product matrix, we take the first row of matrix A and the first column of matrix B, multiply corresponding elements, and add up the products: \(1 \times 12 + (-12) \times (-6) + 4 \times 10\). It's a simple yet methodical operation that is the cornerstone of matrix multiplication.
In essence, you multiply each element of a row in the first matrix with each corresponding element of a column in the second matrix and add up all those products. This is done for each element of the resulting matrix.
For example, if we're calculating the entry in the first row and first column of the product matrix, we take the first row of matrix A and the first column of matrix B, multiply corresponding elements, and add up the products: \(1 \times 12 + (-12) \times (-6) + 4 \times 10\). It's a simple yet methodical operation that is the cornerstone of matrix multiplication.
Graphing Utility
A graphing utility is a powerful tool that can handle a multitude of mathematical operations, including matrix multiplication. It's particularly useful when dealing with large or complex matrices where manual computation would be tedious or prone to error.
When using a graphing utility for matrix operations, you can typically enter the matrices and then select the appropriate function for multiplication. The utility then takes care of the compatibility check and the computation of the dot product for each element of the product matrix.
Our example of multiplying matrices A and B can become a straightforward process with such a tool. It allows for quick verification of results and can be an invaluable aid in confirming the correct solutions for homework or other exercises.
When using a graphing utility for matrix operations, you can typically enter the matrices and then select the appropriate function for multiplication. The utility then takes care of the compatibility check and the computation of the dot product for each element of the product matrix.
Our example of multiplying matrices A and B can become a straightforward process with such a tool. It allows for quick verification of results and can be an invaluable aid in confirming the correct solutions for homework or other exercises.
Matrix Operations
Matrix operations extend beyond just multiplication; they include addition, subtraction, scalar multiplication, transposition, and more. Each operation comes with its own set of rules and applications.
Being able to multiply matrices like A and B is just the beginning. Understanding the different operations can help solve systems of equations, transform geometric figures, and perform data analysis, among other applications. Each operation contributes to a broader mathematical toolbox that students can apply in various fields, such as computer science, physics, and engineering.
With this toolkit, recognizing operations like the dot product in matrix multiplication becomes easier, allowing students to approach problems systematically and with confidence.
Being able to multiply matrices like A and B is just the beginning. Understanding the different operations can help solve systems of equations, transform geometric figures, and perform data analysis, among other applications. Each operation contributes to a broader mathematical toolbox that students can apply in various fields, such as computer science, physics, and engineering.
With this toolkit, recognizing operations like the dot product in matrix multiplication becomes easier, allowing students to approach problems systematically and with confidence.
Other exercises in this chapter
Problem 47
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Find a system of linear equations that has the given solution. (There are many correct answers.) \((a, a+4, a),\) where \(a\) is a real number
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Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form . Use the matrix capabilities of a graphing utility to write t
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