Question

Determine whether the product is defined. If it is defined, find the product; if it is not write "not defined." $$ \left[\begin{array}{rrr} 4 & -2 & 3 \\ 0 & 1 & 2 \\ -1 & 0 & 1 \end{array}\right]\left[\begin{array}{rr} 2 & 6 \\ 1 & -1 \\ 0 & 2 \end{array}\right] $$

Short answer

Defined; Matrix C = \begin{array}{rr}6 & 32 \1 & 3 \-2 & -4d{array}

Step-by-step solution

- Check if the product is defined

To determine if the product of two matrices is defined, check the dimensions of the matrices. The number of columns in the first matrix must equal the number of rows in the second matrix. Matrix A is a 3x3 matrix (3 rows and 3 columns), and Matrix B is a 3x2 matrix (3 rows and 2 columns). Since the number of columns in Matrix A (3) is equal to the number of rows in Matrix B (3), the product is defined.

- Setup the result matrix

The resulting matrix, let's call it Matrix C, will have dimensions based on the rows of Matrix A and the columns of Matrix B. Therefore, Matrix C will be a 3x2 matrix.

- Calculate each element of the resulting matrix

Each element of Matrix C is calculated by taking the dot product of the corresponding row from Matrix A with the corresponding column from Matrix B. Below are the detailed calculations for each element of Matrix C:\[C_{11} = 4*2 + (-2)*1 + 3*0 = 8 - 2 + 0 = 6\]\[C_{12} = 4*6 + (-2)*(-1) + 3*2 = 24 + 2 + 6 = 32\]\[C_{21} = 0*2 + 1*1 + 2*0 = 0 + 1 + 0 = 1\]\[C_{22} = 0*6 + 1*(-1) + 2*2 = 0 - 1 + 4 = 3\]\[C_{31} = (-1)*2 + 0*1 + 1*0 = -2 + 0 + 0 = -2\]\[C_{32} = (-1)*6 + 0*(-1) + 1*2 = -6 + 0 + 2 = -4\]

- Write the resulting matrix

Now that each element of the resulting matrix has been calculated, we can write Matrix C as:\[\begin{array}{rr}6 & 32 \1 & 3 \-2 & -4\end{array}\]

Key concepts

Matrix Dimensions

When working with matrix multiplication, understanding matrix dimensions is crucial. Dimensions are defined by the number of rows and columns a matrix contains. For example, if a matrix has 3 rows and 3 columns, it is referred to as a '3x3' matrix. In our example, Matrix A is a 3x3 matrix and Matrix B is a 3x2 matrix.

To determine if two matrices can be multiplied, you must compare their dimensions. Specifically, the number of columns in the first matrix must equal the number of rows in the second matrix. In this exercise, Matrix A has 3 columns, and Matrix B has 3 rows, so the multiplication is defined.

When the multiplication is defined, the resulting matrix will have dimensions that are a combination of the first matrix's rows and the second matrix’s columns. In this case, the resulting matrix (Matrix C) will be a 3x2 matrix.

Dot Product

The dot product is a fundamental operation in matrix multiplication. It involves multiplying corresponding elements from a row of the first matrix and a column of the second matrix and then summing those products. This gives a single value for each element in the resulting matrix.

Here's a step-by-step method to calculate the dot product:

• Choose a row from the first matrix and a column from the second matrix.
• Multiply each element of the row by the corresponding element in the column.
• Sum all these products to get the dot product result.

For Matrix A and Matrix B, let's calculate the dot product of the first row of Matrix A and the first column of Matrix B: \[C_{11} = 4*2 + (-2)*1 + 3*0 = 8 - 2 + 0 = 6 \]You repeat this process for every combination of rows and columns to fill out the resulting matrix. Each calculation helps determine one element in the new matrix.

Resulting Matrix

After calculating the dot products for all row-column pairs, we construct the resulting matrix. This new matrix contains all the calculated dot products as its elements.

In our example, the resulting matrix (Matrix C) is a 3x2 matrix. Here are the final calculations for each element in Matrix C:

\[C_{11} = 4*2 + (-2)*1 + 3*0 = 6 \] \[C_{12} = 4*6 + (-2)*(-1) + 3*2 = 32 \] \[C_{21} = 0*2 + 1*1 + 2*0 = 1 \] \[C_{22} = 0*6 + 1*(-1) + 2*2 = 3 \] \[C_{31} = (-1)*2 + 0*1 + 1*0 = -2 \] \[C_{32} = (-1)*6 + 0*(-1) + 1*2 = -4 \]
Each of these elements corresponds to a specific position in the resulting matrix:
\[\begin{array}{rr} 6 & 32 \ 1 & 3 \ -2 & -4 \end{array} \]

Thus, Matrix C is constructed from the dot products of the rows of Matrix A and the columns of Matrix B.