Problem 49 - College Algebra

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.(Hint: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{ll}{1} & {0} \\ {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{-2} & {3} & {4} \\ {-1} & {1} & {-5}\end{array}\right], C=\left[\begin{array}{rr}{0.5} & {0.1} \\ {1} & {0.2} \\\ {-0.5} & {0.3}\end{array}\right], D=\left[\begin{array}{rrr}{1} & {0} & {-1} \\ {-6} & {7} & {5} \\ {4} & {2} & {1}\end{array}\right] $$ $$ A(B C) $$

Step-by-Step Solution

Answer

The result is \( \begin{bmatrix} 1.5 & 1.6 \\ 11.5 & -0.1 \end{bmatrix} \).

Step 1: Analyze matrix dimensions for multiplication BC

Matrix \( B \) is a 2x3 matrix, and matrix \( C \) is a 3x2 matrix. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Since B (2x3) and C (3x2) satisfy this condition, the multiplication is possible.

Step 2: Perform the multiplication (BC)

Now we multiply matrix \( B \) with matrix \( C \). The resulting matrix will have dimensions 2x2. The matrix multiplication is performed as follows:\[BC = \begin{bmatrix} -2 & 3 & 4 \ -1 & 1 & -5\end{bmatrix}\begin{bmatrix} 0.5 & 0.1 \ 1 & 0.2 \ -0.5 & 0.3\end{bmatrix} = \begin{bmatrix} (-2)(0.5) + (3)(1) + (4)(-0.5) & (-2)(0.1) + (3)(0.2) + (4)(0.3) \ (-1)(0.5) + (1)(1) + (-5)(-0.5) & (-1)(0.1) + (1)(0.2) + (-5)(0.3)\end{bmatrix}\]Solving each element, we get:\[BC = \begin{bmatrix} 1.5 & 1.6 \ 3 & -1.1\end{bmatrix}\]

Step 3: Analyze matrix dimensions for multiplication A(BC)

Matrix \( A \) is a 2x2 matrix, and the result of \( BC \) is also a 2x2 matrix. Since the dimensions are compatible (2 columns of A and 2 rows of BC), the multiplication \( A(BC) \) is possible.

Step 4: Perform the multiplication A(BC)

Now multiply matrix \( A \) with the result of \( BC \):\[A(BC) = \begin{bmatrix} 1 & 0 \ 2 & 3\end{bmatrix}\begin{bmatrix} 1.5 & 1.6 \ 3 & -1.1\end{bmatrix} = \begin{bmatrix} (1)(1.5) + (0)(3) & (1)(1.6) + (0)(-1.1) \ (2)(1.5) + (3)(3) & (2)(1.6) + (3)(-1.1)\end{bmatrix}\]Solving each element, we get:\[A(BC) = \begin{bmatrix} 1.5 & 1.6 \ 11.5 & -0.1\end{bmatrix}\]

Step 5: Conclusion: Result of operation

The result of the matrix operation \( A(BC) \) is:\[\begin{bmatrix} 1.5 & 1.6 \ 11.5 & -0.1\end{bmatrix}\].

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial in understanding the structure and size of matrices. Every matrix is defined by two measurements: the number of rows and the number of columns it contains. Together, these numbers create what's known as the "dimensions" of a matrix. For example, a matrix with 2 rows and 3 columns has dimensions 2x3.

When performing operations like matrix multiplication, understanding dimensions is essential. This is because the ability to multiply two matrices depends on specific dimension compatibility, which we'll discuss next. Analyzing dimensions before proceeding with any operations ensures that the operation is mathematically valid, saving time and preventing errors.

When dealing with exercises involving matrices, always note the dimensions first. This will guide you on what's possible and what's not in your matrix operations journey.

Matrix Compatibility

The compatibility of matrices is a fundamental concept in matrix algebra, especially when performing multiplication. To multiply two matrices, we must verify their dimensions to ensure compatibility. Specifically, the number of columns in the first matrix must equal the number of rows in the second matrix. This requirement is crucial for the multiplication to be possible.

For example, if matrix A has dimensions 2x3, it can only be multiplied with a matrix B that has 3 rows, regardless of how many columns B has. In our original exercise, matrix B (2x3) and matrix C (3x2) were compatible for multiplication because B's columns matched C's rows.

• A 2x3 matrix can be multiplied with a 3x2 matrix.
• A 3x3 matrix cannot be multiplied with a 2x3 matrix.

This framework remains consistent across all matrix multiplications, making it a reliable tool for determining matrix operations.

Matrix Algebra

Matrix algebra involves various rules and operations to manipulate matrices. Basic operations include addition, subtraction, and multiplication, each following specific rules of algebra. Unlike simple arithmetic, matrices require adherence to their unique rules, especially regarding their dimensions and compatibility.

In addition to these operations, matrix algebra also enables transformations such as finding inverses, determinants, or transposes of matrices, though these were not needed in our specific exercise.

• Addition and subtraction must occur between matrices of the same dimensions.
• Multiplication has different rules which involve compatibility as discussed.

Understanding these principles allows for more complex problem-solving and applications in areas such as computer graphics and data science.

Matrix Operations

Matrix operations involve manipulating matrices through addition, subtraction, and multiplication. In our exercise, we focused on multiplication, a slightly more complex operation foundational in matrix algebra.

When performing matrix multiplication, each element in the resulting matrix is calculated by summing the products of corresponding elements from the rows of the first matrix and columns of the second matrix.

• Ensure the matrices are compatible by checking their dimensions.
• Calculate the resulting matrix's elements carefully, following the multiplication and summation rules.
• Verify each step to avoid calculation errors.

Understanding how matrix operations are structured will greatly aid in executing such calculations accurately and efficiently, leading to precise results critical in various mathematical applications.