Problem 24 - Algebra and Trigonometry

Question

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right] $$ $$ D=\left[\begin{array}{ll}{7} & {3}\end{array}\right] \quad E=\left[\begin{array}{l}{1} \\ {2} \\ {0}\end{array}\right] \quad F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$ $$ G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right] \quad H=\left[\begin{array}{rr}{3} & {1} \\ {2} & {-1}\end{array}\right] $$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ \begin{array}{ll}{\text { (a) } 3 B+2 C} & {\text { (b) } 2 H+D}\end{array} $$

Step-by-Step Solution

Answer

3B + 2C can be performed; 2H + D cannot be performed.

Step 1: Determine the dimensions of matrices for operation (a)

Matrix \( B \) has dimensions \( 2 \times 3 \) and matrix \( C \) also has dimensions \( 2 \times 3 \). Both matrices must have the same dimensions to add them, which they do. Therefore, the operation \( 3B + 2C \) is feasible.

Step 2: Scale matrices for operation (a)

First, multiply each element of matrix \( B \) by 3:\[3B = \left[\begin{array}{ccc}{9} & {\frac{3}{2}} & {15} \ {3} & {-3} & {9}\end{array}\right]\]Next, multiply each element of matrix \( C \) by 2:\[2C = \left[\begin{array}{ccc}{4} & {-5} & {0} \ {0} & {4} & {-6} \end{array}\right]\]

Step 3: Add matrices for operation (a)

Sum the corresponding elements of \( 3B \) and \( 2C \):\[3B + 2C = \left[\begin{array}{ccc}{9 + 4} & {\frac{3}{2} - 5} & {15 + 0} \ {3 + 0} & {-3 + 4} & {9 - 6}\end{array}\right] = \left[\begin{array}{ccc}{13} & {-\frac{7}{2}} & {15} \ {3} & {1} & {3}\end{array}\right]\]

Step 4: Determine the dimensions for operation (b)

Matrix \( H \) has dimensions \( 2 \times 2 \) and matrix \( D \) has dimensions \( 1 \times 2 \). To add two matrices, they must have the same dimensions, which they do not in this case. Therefore, the operation \( 2H + D \) is not feasible. However, let's redefine and explore the addition attempt if possible by breaking them down.

Step 5: Scale matrix for operation (b)

Multiply each element of matrix \( H \) by 2:\[2H = \left[\begin{array}{cc}{6} & {2} \ {4} & {-2}\end{array}\right]\]

Step 6: Attempt to add matrices for operation (b)

Matrix \( D \) is a row vector \( \left[\begin{array}{cc}{7} & {3}\end{array}\right] \) of size \( 1 \times 2 \), and 2H is a matrix of size \( 2 \times 2 \). These dimensions are incompatible for addition, so the operation \( 2H + D \) cannot be performed.

Key Concepts

Matrix Addition

Matrix addition is an operation where two matrices are added together by adding their corresponding elements. For this operation to be possible, the matrices must have the same dimensions, meaning they must have the same number of rows and columns.
For example, consider matrices \(B\) and \(C\), both having dimensions \(2 \times 3\). They can be added because:

• They both have 2 rows.
• They both have 3 columns.

The addition of the scaled matrices 3B and 2C involves adding each element from a position in 3B to the corresponding element in 2C, resulting in a new matrix.
If matrices of different dimensions, like matrix \(H\) \((2 \times 2)\) and matrix \(D\) \((1 \times 2)\), are attempted to be added, the operation is not feasible. This is because the number of elements across corresponding positions does not match, making addition impossible without adjusting the matrices to the same shape.

Matrix Scaling

Matrix scaling involves multiplying all elements of a matrix by a scalar value. This operation is quite simple as it affects each element uniformly across the matrix.
In our example, when scaling matrix \(B\) by 3, each element of \(B\) is multiplied by 3, so a matrix element like 3 becomes 9, and \(\frac{1}{2}\) becomes \(\frac{3}{2}\). Similarly, scaling matrix \(C\) by 2 means each element in \(C\) is doubled.
This type of scaling is necessary as a preparatory step before matrix addition in cases where matrices need operations like \(3B + 2C\). By scaling matrices individually by their respective factors, you ensure they are ready for further operations to be performed accurately. Scaling is especially useful in situations where elements of the matrix represent weighted quantities, such as finding the resultant effect of different magnitudes represented in the matrix.

Matrix Dimensions

Understanding matrix dimensions is crucial in performing any matrix operations accurately. The dimensions of a matrix are defined by the number of rows it has and the number of its columns, typically expressed in the form \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.
Knowing the dimensions of matrices is particularly important for operations such as addition and subtraction, where matrices must have identical dimensions to proceed. For instance, matrices \(B\) and \(C\), with dimensions \(2 \times 3\), can be added without issue.
However, while attempting to add matrix \(H\) \((2 \times 2\)) with matrix \(D\) \((1 \times 2\)), you'll find that the operation fails because their dimensions do not align. Understanding dimensions helps quickly identify possible or impossible operations, guiding the setup for complex algebraic calculations.