Problem 4
Question
Fill in the missing entries in the product matrix. $$\left[\begin{array}{rrr} 3 & 1 & 2 \\ -1 & 2 & 0 \\ 1 & 3 & -2 \end{array}\right]\left[\begin{array}{rrr} -1 & 3 & -2 \\ 3 & -2 & -1 \\ 2 & 1 & 0 \end{array}\right]=\left[\begin{array}{rrr} 4 & \mathbb{I} & -7 \\ 7 & -7 & \mathbb{I} \\ -5 & -5 & -5 \end{array}\right]$$
Step-by-Step Solution
Verified Answer
The missing entries are 9 and 0.
1Step 1: Understand Matrix Multiplication
The product of two matrices is calculated by taking the dot product of rows of the first matrix with columns of the second matrix. The element in the resulting matrix at position (i, j) is equal to the sum of products of corresponding elements.
2Step 2: Calculate Element at Position (1,2)
For the element in row 1 and column 2, take row 1 of the first matrix and column 2 of the second matrix: \[ (3 \cdot 3) + (1 \cdot -2) + (2 \cdot 1) = 9 - 2 + 2 = 9.\] So, the element in position (1,2) is 9.
3Step 3: Calculate Element at Position (2,3)
For the element in row 2 and column 3, take row 2 of the first matrix and column 3 of the second matrix: \[ (-1 \cdot -2) + (2 \cdot -1) + (0 \cdot 0) = 2 - 2 + 0 = 0.\] So, the element in position (2,3) is 0.
4Step 4: Complete the Product Matrix
By filling in the previously missing elements, our complete product matrix becomes: \[ \begin{bmatrix} 4 & 9 & -7 \ 7 & -7 & 0 \ -5 & -5 & -5 \end{bmatrix}\]
Key Concepts
Dot ProductProduct MatrixMatrix Operations
Dot Product
The dot product is a fundamental operation in matrix multiplication. It involves multiplying corresponding elements of a row from the first matrix with a column from the second matrix.
Then, you add these products together to get a single number, which becomes an element in the resulting product matrix.
Understanding how the dot product works is crucial for mastering matrix operations.
For instance, for the first element in position (1,2), use the dot product of the first row of the first matrix and the second column of the second matrix:
Then, you add these products together to get a single number, which becomes an element in the resulting product matrix.
Understanding how the dot product works is crucial for mastering matrix operations.
For instance, for the first element in position (1,2), use the dot product of the first row of the first matrix and the second column of the second matrix:
- Multiply: \(3 \times 3 = 9\)
- Multiply: \(1 \times -2 = -2\)
- Multiply: \(2 \times 1 = 2\)
- Add all the products: \(9 - 2 + 2 = 9\)
Product Matrix
The product matrix is the result of multiplying two matrices. It's formed by calculating the dot product for each pair of row and column from the respective matrices.
Each element in the product matrix corresponds to the interaction of a particular row from the first matrix and a column from the second matrix.
The number of rows in the product matrix equals the number of rows in the first matrix.
The number of columns equals the number of columns in the second matrix.
In our example, after filling in the missing entries using calculated dot products, the product matrix is:\[\begin{bmatrix} 4 & 9 & -7 \7 & -7 & 0 \-5 & -5 & -5 \end{bmatrix}\]This illustrates how each element in the product matrix originates from combining rows and columns.
Each element in the product matrix corresponds to the interaction of a particular row from the first matrix and a column from the second matrix.
The number of rows in the product matrix equals the number of rows in the first matrix.
The number of columns equals the number of columns in the second matrix.
In our example, after filling in the missing entries using calculated dot products, the product matrix is:\[\begin{bmatrix} 4 & 9 & -7 \7 & -7 & 0 \-5 & -5 & -5 \end{bmatrix}\]This illustrates how each element in the product matrix originates from combining rows and columns.
Matrix Operations
Matrix operations involve various methods to handle matrices, such as addition, subtraction, and multiplication, which is our focus here.
Matrix multiplication is not the same as multiplying real numbers; it requires specific rules and operations. When performing matrix multiplication, ensure that the number of columns in the first matrix equals the number of rows in the second matrix.
Otherwise, the operation is undefined
Use the following steps for matrix multiplication:
By understanding these concepts, students can solve complex mathematical and real-world problems.
Matrix multiplication is not the same as multiplying real numbers; it requires specific rules and operations. When performing matrix multiplication, ensure that the number of columns in the first matrix equals the number of rows in the second matrix.
Otherwise, the operation is undefined
Use the following steps for matrix multiplication:
- Select a row from the first matrix and a column from the second matrix.
- Calculate the dot product by multiplying corresponding elements and summing them up.
- Place the resulting sum in the corresponding position of the product matrix.
By understanding these concepts, students can solve complex mathematical and real-world problems.
Other exercises in this chapter
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