Problem 29 - 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. $$ \begin{array}{ll}{\text { (a) } G F} & {\text { (b) } G E}\end{array} $$

Step-by-Step Solution

Answer

(a) GF = G; (b) GE = \([ -1, 8, -1 ]^T\).

Step 1: Determine compatibility for Operation (a) GF

First, check if the matrices \(G\) and \(F\) can be multiplied. Matrix \(G\) is a 3x3 matrix and matrix \(F\) is also a 3x3 matrix. Since the number of columns in \(G\) equals the number of rows in \(F\), the multiplication \(GF\) is permissible.

Step 2: Perform Matrix Multiplication for GF

Multiply the matrices \(G\) and \(F\). To find each element of the resulting matrix, take the dot product of the corresponding row from \(G\) with the corresponding column from \(F\). Since \(F\) is the identity matrix, \(GF\) will result in \(G\) itself:\[GF = \left[ \begin{array}{ccc} 5 & -3 & 10 \ 6 & 1 & 0 \ -5 & 2 & 2 \end{array} \right]\].

Step 3: Determine compatibility for Operation (b) GE

Check if matrices \(G\) and \(E\) can be multiplied. Matrix \(G\) is a 3x3 matrix and matrix \(E\) is a 3x1 matrix. Since the number of columns in \(G\) equals the number of rows in \(E\), the multiplication \(GE\) is possible.

Step 4: Perform Matrix Multiplication for GE

Multiply matrices \(G\) and \(E\). The dimensions of the resulting matrix will be determined by the rows of \(G\) and columns of \(E\). Compute each element by taking the dot product of the corresponding row of \(G\) with the column of \(E\):\[GE = \left[ \begin{array}{c} (5\cdot 1) + (-3\cdot 2) + (10\cdot 0) \ (6\cdot 1) + (1\cdot 2) + (0\cdot 0) \ (-5\cdot 1) + (2\cdot 2) + (2\cdot 0) \end{array} \right] = \left[ \begin{array}{c} -1 \ 8 \ -1 \end{array} \right]\].

Key Concepts

Matrix Compatibility

Matrix compatibility is a fundamental concept in matrix multiplication. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. This is because each element of the resulting matrix is computed by taking the dot product of a row from the first matrix and a column from the second matrix.
For example, consider multiplying matrix \(G\), which is a 3x3 matrix, with matrix \(F\), another 3x3 matrix. Since they have equal dimensions (both have 3 columns and 3 rows), they are compatible for multiplication. Similarly, for matrix \(G\) and matrix \(E\), where \(E\) is a 3x1 matrix, compatibility is ensured because the number of columns in \(G\) (3) matches the number of rows in \(E\) (3).
When matrices are not compatible, matrix multiplication cannot be performed. Understanding matrix compatibility is crucial as it governs whether an operation is possible.

Identity Matrix

An identity matrix is a special kind of matrix that plays a significant role in matrix operations. It is a square matrix with ones on the main diagonal and zeros elsewhere. For a matrix to be an identity matrix, it must have the same number of rows and columns, making it a square matrix.
The defining property of the identity matrix is that when any matrix is multiplied by it, the original matrix remains unchanged. This is akin to the number one in multiplication of numbers. For example, if we multiply any 3x3 matrix by a 3x3 identity matrix, the result will be the original matrix itself.
In the exercise, multiplying \(G\) by the identity matrix \(F\) resulted in \(GF = G\). This confirms the identity property: multiplying any matrix by the identity matrix results in the matrix itself.

Dot Product

The dot product is a key mathematical operation used in matrix multiplication. It involves multiplying corresponding elements from a row of the first matrix and a column of the second matrix, then summing those products. The sum is a single number that forms an element of the resulting matrix.
To compute a dot product, follow these steps:

• Multiply each element of the row of the first matrix by the corresponding element in the column of the second matrix.
• Add the results of these multiplications together.

For example, in computing the element in the first row and first column of the result of \(GE\), take the first row of \(G\) \[ (5, -3, 10) \] and the only column of \(E\) \[ (1, 2, 0) \]. Perform the dot product: \(5 \cdot 1 + (-3) \cdot 2 + 10 \cdot 0 = -1\).
This simple yet powerful operation underpins all of matrix multiplication, connecting individual elements of matrices to form a whole.