Problem 28 - 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. $$ \begin{array}{ll}{\text { (a) } B C} & {\text { (b) } B F}\end{array} $$

Step-by-Step Solution

Answer

(a) BC is not defined, (b) BF = B

Step 1: Verify Matrix Multiplication for BC

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Matrix \(B\) is a 2x3 matrix, and matrix \(C\) is also a 2x3 matrix. For multiplication \(BC\) to be possible, \(B\) must have the same number of columns as the rows of \(C\), but \(B\) has 3 columns and \(C\) has 2 rows, thus multiplication \(BC\) is not defined.

Step 2: Verify Matrix Multiplication for BF

Matrix \(B\) is a 2x3 matrix, and matrix \(F\) is a 3x3 matrix. The multiplication \(BF\) is possible since the number of columns in \(B\) (3) is equal to the number of rows in \(F\) (3). The resulting matrix will be a 2x3 matrix.

Step 3: Perform Matrix Multiplication BF

We calculate the product \(BF\) by multiplying each row of \(B\) with each column of \(F\). Since \(F\) is an identity matrix, \(BF\) will result in the same matrix \(B\). Thus, \[BF = \left[ \begin{array}{ccc} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{array} \right]\].

Key Concepts

Matrix Dimensions

Matrix dimensions are an essential aspect of understanding how matrices operate, particularly when dealing with matrix multiplication.
When we talk about the dimensions of a matrix, we are referring to its size, which is given by the number of rows by the number of columns. For instance, Matrix A in the above example is a 2x2 matrix, meaning it has 2 rows and 2 columns.
Understanding matrix dimensions is crucial because it determines the viability of operations such as matrix multiplication.

• To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
• For example, consider matrix B which is a 2x3 matrix and matrix C which is also a 2x3 matrix. Given their dimensions, multiplication cannot occur because the number of columns in B (3) is not the same as the number of rows in C (2).

Dimensions also determine the dimensions of the resulting matrix from multiplication. For example, if multiplying a 2x3 matrix with a 3x3 matrix, like matrix B and F in our example, the result is a 2x3 matrix.

Matrix Product

Matrix multiplication, also known as the matrix product, is a fundamental concept in linear algebra that combines two matrices to form a new matrix.
The process involves multiplying the elements of rows from the first matrix by the elements of columns from the second matrix and summing the results. It is essential that the number of columns in the first matrix matches the number of rows in the second matrix, allowing the operation to proceed.

• Performing the operation involves taking each element in the row of the first matrix and multiplying it by the corresponding element in the column of the second matrix, adding these products.
• For example, if matrix B is multiplied by matrix F, each entry in the resulting matrix corresponds to the summation of products of rows from B and columns from F.

The matrix product's usefulness cannot be overstated, with applications spanning from solving systems of linear equations to transformations in computer graphics.

Identity Matrix

The identity matrix is a special type of matrix that plays an important role in matrix operations. It is the equivalent of the number '1' in matrix multiplication.
It is a square matrix meaning it has the same number of rows and columns, and it is filled with ones on the diagonal from top left to bottom right and zeros elsewhere. When any matrix is multiplied by an identity matrix, it remains unchanged, similar to how any number remains the same when multiplied by 1.

• An identity matrix, when applied in matrix multiplication, serves as a neutral element. For instance, in the multiplication of matrix B by matrix F, matrix F being an identity matrix ensures that the result will be the same as matrix B.
• The importance of the identity matrix extends to solving linear systems and finding the inverse of a matrix, where it serves as the basis or result for achieving such solutions.

Thus, the identity matrix is essential for preserving properties of original matrices during multiplication, acting as a fundamental building block in linear algebra.