Problem 28 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The result of \(GF\) is the same as \(G\), since \(F\) is an identity matrix.

Step 1: Understand Matrix Multiplication

Matrix multiplication is possible between an \(m \times n\) matrix and an \(n \times p\) matrix, resulting in an \(m \times p\) matrix. The number of columns in the first matrix must equal the number of rows in the second matrix.

Step 2: Identify Dimensions of Matrix G

Matrix \(G\) is given as \(3 \times 3\), meaning it has 3 rows and 3 columns.

Step 3: Identify Dimensions of Matrix F

Matrix \(F\) is an identity matrix of dimensions \(3 \times 3\), meaning it also has 3 rows and 3 columns.

Step 4: Check Matrix Multiplication Compatibility

Since both matrices \(G\) and \(F\) are \(3 \times 3\), matrix multiplication is possible because the number of columns in \(G\) matches the number of rows in \(F\).

Step 5: Perform Matrix Multiplication

To find \(GF\), multiply each row of \(G\) by each column of \(F\):Since \(F\) is an identity matrix, multiplying any matrix by \(F\) leaves the matrix unchanged. Therefore, \(GF = G\). \[ G \times F = \begin{bmatrix} 5 & -3 & 10 \ 6 & 1 & 0 \ -5 & 2 & 2 \end{bmatrix} \]

Step 6: Confirm the Result

Since multiplying by an identity matrix does not change the original matrix, we confirm the result as \(GF = G\).

Key Concepts

Identity Matrix

An identity matrix is a special type of square matrix that acts like the number 1 in matrix multiplication. It is a diagonal matrix where all the elements on the main diagonal are 1, and all off-diagonal elements are 0.
The identity matrix plays a key role in maintaining the identity of another matrix when multiplied by it.

• Multiplying any matrix by an identity matrix will result in the original matrix itself.
• For instance, if you have a matrix \(M\) and you multiply it by an identity matrix, \(M \times I = M\).

Thus, the identity matrix ensures that the multiplication doesn't alter the original structure of the matrix. In this problem, matrix \(F\) is an identity matrix, which means when we multiply \(G\) by \(F\), \(G\) remains unchanged.

Matrix Dimensions

Matrix dimensions tell us the size and structure of a matrix. They are represented as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. Understanding dimensions is crucial for determining if matrices can be multiplied.

• A matrix with dimensions \(3 \times 3\) means it has 3 rows and 3 columns.
• In this exercise, both matrices \(G\) and \(F\) are \(3 \times 3\).

Recognizing matrix dimensions helps in evaluating matrix compatibility for multiplication and predicting the dimensions of the resulting matrix. The resulting product matrix from multiplying \(G\) and \(F\) (both \(3 \times 3\)) would also be a \(3 \times 3\) matrix.

Matrix Compatibility Check

Checking matrix compatibility is the first step before performing matrix multiplication. For two matrices to be compatible for multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.

• For instance, a matrix \(A\) with dimensions \(m \times n\) can be multiplied with a matrix \(B\) of dimensions \(n \times p\) to get a resulting \(m \times p\) matrix.
• This ensures that each element of the resulting matrix is derived from valid direct multiplication and summation of corresponding elements.

In this problem, both matrices \(G\) and \(F\) are \(3 \times 3\), making them compatible for multiplication. Once compatibility is confirmed, the multiplication process can proceed smoothly, ensuring correct results.