Problem 19 - College Algebra

Question

Use the matrices below to perform matrix multiplication. \(A=\left[\begin{array}{rr}-1 & 5 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{rrr}3 & 6 & 4 \\ -8 & 0 & 12\end{array}\right], C=\left[\begin{array}{rr}4 & 10 \\ -2 & 6 \\ 5 & 9\end{array}\right], D=\left[\begin{array}{rrr}2 & -3 & 12 \\ 9 & 3 & 1 \\ 0 & 8 & -10\end{array}\right]\) \(B C\)

Step-by-Step Solution

Answer

The product \( BC \) is \( \begin{bmatrix} 20 & 102 \\ 28 & 28 \end{bmatrix} \)."

Step 1: Understand Matrix Sizes for Multiplication

Before we perform matrix multiplication, we need to determine if the multiplication is possible by checking the dimensions of the matrices. Matrix \( B \) is a \( 2 \times 3 \) matrix, and matrix \( C \) is a \( 3 \times 2 \) matrix. Since the number of columns in \( B \) matches the number of rows in \( C \), the multiplication \( BC \) is possible and will result in a \( 2 \times 2 \) matrix.

Step 2: Multiply Matrices B and C

We will calculate each element of the resulting \( 2 \times 2 \) matrix by taking the dot product of the corresponding row of \( B \) and column of \( C \).**Element (1,1):** \[ (3 \times 4) + (6 \times -2) + (4 \times 5) = 12 - 12 + 20 = 20 \]**Element (1,2):** \[ (3 \times 10) + (6 \times 6) + (4 \times 9) = 30 + 36 + 36 = 102 \]**Element (2,1):** \[ (-8 \times 4) + (0 \times -2) + (12 \times 5) = -32 + 0 + 60 = 28 \]**Element (2,2):** \[ (-8 \times 10) + (0 \times 6) + (12 \times 9) = -80 + 0 + 108 = 28 \]

Step 3: Write the Resulting Product Matrix

Using the calculated elements from Step 2, we can write the resulting matrix \( BC \) as follows:\[ BC = \begin{bmatrix} 20 & 102 \ 28 & 28 \end{bmatrix} \]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in matrix multiplication. Each matrix has a specific shape, determined by the number of rows and columns it contains. For example, if a matrix has 2 rows and 3 columns, it is referred to as a "2 x 3" matrix. The order of multiplication matters, and the rule of dimension compatibility must be followed. This rule states that the number of columns in the first matrix must equal the number of rows in the second matrix. Only then is multiplication possible. If we take our matrices from the exercise, matrix \( B \) has dimensions of 2 x 3, while matrix \( C \) is 3 x 2. Because the number of columns in \( B \) (3) aligns perfectly with the number of rows in \( C \) (also 3), these matrices can be multiplied. The resulting matrix after multiplication will take on new dimensions, specifically 2 x 2. Remember, these new dimensions are a combination of the outer dimensions of the multiplied matrices, where the result takes the row count of the first and the column count of the second matrix.

Dot Product

The dot product is a mathematical operation that plays a pivotal role in calculating the elements of the resultant matrix during multiplication. When working with matrices, the dot product helps to identify the values inside each entry of the resultant matrix. Here's how it works: To find each element in the resulting matrix from multiplying two matrices \( B \) and \( C \), you take the corresponding row from \( B \) and column from \( C \), and then compute their dot product. This involves multiplying the elements of the row and column, one by one, and then summing these products. For instance, to find the upper left entry of the result, you would multiply each element of the first row of \( B \) with each corresponding element of the first column of \( C \) and add them all together. So, for element (1,1) of the resulting matrix, the computation would be \( (3 \times 4) + (6 \times -2) + (4 \times 5) \), which equals 20.

Matrix Result

The result of a matrix multiplication operation is a new matrix. In our exercise, performing the multiplication \( B \times C \) gives a resultant matrix with dimensions defined by the outer dimensions of the original matrices, thus a 2 x 2 matrix. Calculating each element of this matrix involves using the dot product of rows from the first matrix and columns from the second, as detailed in the previous section. For clarity, although this process involves several calculations, each element needs careful attention. In our specific case:

• For element (1,1), the result is 20 after calculation.
• Element (1,2) is determined to be 102.
• The (2,1) position holds the value 28.
• Finally, (2,2) also comes out to be 28, after solving the dot products precisely.

Rendering the final matrix, we have:\[BC = \begin{bmatrix} 20 & 102 \ 28 & 28 \end{bmatrix}\]This matrix effectively demonstrates the transformation that occurs via multiplication.

Matrix Arithmetic

Matrix arithmetic refers to operations such as addition, subtraction, and multiplication applied to matrices. Each of these operations has particular rules. In the context of multiplication, matrices must adhere to the compatibility rule discussed in the matrix dimensions section. The arithmetic process itself contains a sequence of systematic steps:

• Checking dimensional compatibility.
• Computing the dot products to form each entry in the resultant matrix.
• Assembling all computed entries into the new matrix.

This systematic structure and methodology make matrix arithmetic both powerful for calculations in areas such as physics and economics, and foundational for more complex mathematical concepts. The use of matrix arithmetic is seen widely in solving systems of equations, transforming geometric figures, and in many computer algorithms, making it an essential technique in mathematics.