Problem 30 - 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. $$ \begin{array}{ll}{\text { (a) } B^{2}} & {\text { (b) } F^{2}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(B^2\) is not defined; (b) \(F^2 = F\).

Step 1: Determine if Matrix Multiplication is Possible for B^2

Matrix multiplication requires that the number of columns in the first matrix be equal to the number of rows in the second matrix. Matrix \( B \) is a \(2 \times 3\) matrix: \[ B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \ {1} & {-1} & {3}\end{array}\right] \] \(B^2\) is defined as \( B \times B \), and since the number of columns in \(B\) (3) does not match the number of rows in \(B\) (2), \(B^2\) is not defined.

Step 2: Calculation of F^2

Matrix \( F \) is a \( 3 \times 3 \) identity matrix: \[ F = \left[ \begin{array}{lll}{1} & {0} & {0} \ {0} & {1} & {0} \ {0} & {0} & {1} \end{array} \right] \] An identity matrix multiplied by itself (or any other compatible matrix) results in the same identity matrix. Therefore, \( F^2 \) is: \[ F^2 = F \times F = \left[ \begin{array}{lll}{1} & {0} & {0} \ {0} & {1} & {0} \ {0} & {0} & {1} \end{array} \right] \]

Key Concepts

Matrix Multiplication

Matrix multiplication is a key operation in matrix algebra, allowing us to combine two matrices to form a new matrix. For two matrices to be compatible for multiplication, the number of columns of the first matrix must be equal to the number of rows of the second one. This dimension matching is essential because it ensures each element from a row in the first matrix can be properly paired and multiplied with an element from a column in the second matrix.

When these conditions are met, you can calculate the product by taking an element from the row of the first matrix and an element from the column of the second matrix, multiplying them, and summing these products. The result is a new element in the resulting matrix. Keep in mind, the resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Let's illustrate this with an example: if matrix A is of size \( m \times n \) and matrix B is of size \( n \times p \), their product AB will be a matrix of size \( m \times p \). If these dimension conditions aren't met, as with \( B^2 \) in our original exercise involving matrix B\( (2 \times 3) \), the multiplication cannot be performed. This highlights the importance of always checking matrix dimensions first.

Identity Matrix

An identity matrix is a special type of square matrix that plays the same role as the number 1 in arithmetic. In matrix terms, multiplying any matrix by an identity matrix (\( I \)) of compatible size will leave the original matrix unchanged. This property is why it's called an "identity" matrix; it preserves the identity of the matrix it multiplies.

Identity matrices are square, meaning they have the same number of rows and columns, and they have 1s on their main diagonal while all other positions contain 0s. For example, our matrix \( F \) from the exercise is a \( 3 \times 3 \) identity matrix:

• \[F = \begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{bmatrix}\]

Multiplying any \( 3 \times 3 \) matrix by this identity matrix \( F \) will return the original matrix.

• \( F \times G = G \)
• \( G \times F = G \)

Furthermore, since an identity matrix does not change a matrix it's multiplied by, multiplying an identity matrix by itself results in the same identity matrix, just as shown in our exercise with \( F^2 = F \times F \), which gives back \( F \).

Matrix Dimensions

Understanding and correctly identifying matrix dimensions is crucial before performing operations like multiplication. Dimensions tell you how many rows and columns a matrix has, often noted as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.

The dimensions of a matrix not only determine its shape but also how and whether it can engage in operations like multiplication with other matrices. For instance, in our exercise, matrix \( B \) was a \( 2 \times 3 \) matrix. For matrix multiplication to be possible with B and itself (attempting \( B^2 \)), \( B \) would need to be a square matrix (i.e., equal rows and columns), but it isn't. Its incompatible dimensions of rows (2) and columns (3) make \( B^2 \) impossible.

Thus, always start by examining and comparing matrix dimensions before proceeding with operations. This way, you can quickly determine what operations are feasible. For operations involving non-square matrices or different dimensions, pay special attention to ensure compatibility.