Problem 36 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( F^2 = F \); it remains an identity matrix.

Step 1: Identify Matrix F

Matrix \( F \) is given as the identity matrix of size 3x3: \[ F = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]. The identity matrix has 1s on the diagonal and 0s elsewhere.

Step 2: Understand Matrix Power

The operation \( F^2 \) means multiplying matrix \( F \) by itself, which is \( F \times F \). For matrices, \( A^2 = A \times A \) when \( A \) is a square matrix.

Step 3: Multiply Matrix F by Itself

To find \( F^2 \):\[F^2 = F \times F = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\] Perform the multiplication of each element considering identity matrix properties.

Step 4: Compute the Product

The identity matrix, when multiplied by itself or any other matrix, results in the same matrix. So, the result is:\[F^2 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]

Step 5: Verify the Result

Since \( F \) is an identity matrix, multiplying it by itself yields the same identity matrix. The calculation confirms this as \( F^2 = F \).

Key Concepts

Identity Matrix

An identity matrix is an essential concept in matrix algebra. It is a special type of square matrix that serves as the 'neutral' element in matrix multiplication. The identity matrix is defined by having 1s along its main diagonal (from top-left to bottom-right) and 0s in all other positions. For example, a 3x3 identity matrix is represented as: \[ I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \] This matrix acts like the number 1 in multiplication of real numbers. When you multiply any matrix by this identity matrix, the original matrix remains unchanged.

• If matrix \( A \) is any matrix that is compatible in size, then \( A \times I = A \) and \( I \times A = A \).
• Identity matrices can be of different sizes such as 2x2, 3x3, and so on, depending on the context or the matrices they are interacting with.

Understanding identity matrices helps in simplifying computations, particularly when dealing with powers of matrices and solving matrix equations.

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra that combines two matrices to form a new matrix. Multiplying matrices involves a systematic combination of rows from the first matrix and columns from the second. This operation is not as simple as multiplying individual elements, as it requires computing the sum of the products. For instance, if you have two matrices \( A \) and \( B \) with dimensions that are compatible (if \( A \) is a \( m \times n \) matrix, then \( B \) must be an \( n \times k \) matrix), the resulting product will be a \( m \times k \) matrix.

• The element in the \( i^{th} \) row and \( j^{th} \) column of the resulting matrix is calculated as the dot product of the \( i^{th} \) row of matrix \( A \) and the \( j^{th} \) column of matrix \( B \).
• This requires summing the products of corresponding elements across the row and the column.

Matrix multiplication is associative and distributive over addition, but it's not commutative (i.e., \( A \times B \) may not equal \( B \times A \)). This characteristic is crucial when dealing with transformations and in solving systems of linear equations.

Square Matrix

A square matrix is one in which the number of rows equals the number of columns. These matrices are significant because many properties and operations in linear algebra apply specifically to them, such as determinant calculations, matrix inversions, and eigenvalues. In practical terms, if a matrix \( A \) is called a square matrix and it has \( n \) rows and columns, it is expressed as an \( n \times n \) matrix.

• Square matrices are the only kinds of matrices that can have a determinant and an inverse (if it is a non-singular matrix).
• They can also be used to describe transformations that preserve dimensions, like rotations and scaling in engineering and computer science.
• They are pivotal in defining concepts of eigenvectors and eigenvalues, which are key in systems analysis and other advanced applications.

Understanding square matrices is fundamental for students learning matrix algebra, as many core linear algebra concepts rely on the properties of square matrices.

Matrix Power

Matrix power is an operation where a square matrix is multiplied by itself a defined number of times. It is only applicable to square matrices. We denote the matrix power of a matrix \( A \) raised to the power of \( n \) as \( A^n \).

• If \( n = 2 \), it means \( A \times A \), often referred to as squaring the matrix.
• If \( n = 3 \), it implies \( A \times A \times A \), and so on.
• The power \( A^0 \) is defined as the identity matrix of the same dimensions as \( A \).

The matrix power of an identity matrix is especially straightforward. For any identity matrix \( I \), no matter how many times you multiply it by itself, the result is still \( I \). This property is what makes the identity matrix unique in matrix algebra. Exploring matrix powers is important for understanding iterative processes and transformations modeled by matrices in various scientific fields.