Problem 32 - 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. $$ A^{2} $$

Step-by-Step Solution

Answer

The square of matrix \(A\) is \(\left[\begin{array}{rr}{4} & {-45} \\ {0} & {49}\end{array}\right]\).

Step 1: Check if multiplying matrix A is possible

Matrix multiplication is possible if the number of columns in the first matrix is equal to the number of rows in the second matrix. Matrix \(A\) is a \(2 \times 2\) matrix. When multiplying \(A\) by itself, this condition is satisfied because both matrices have dimensions \(2 \times 2\).

Step 2: Write out the expression for A squared

The expression for \(A^2\) is given by multiplying matrix \(A\) by itself: \[A^2 = A \times A.\]

Step 3: Perform matrix multiplication

To multiply two \(2 \times 2\) matrices, calculate each element of the resulting matrix by summing the products of the corresponding row elements of the first matrix and column elements of the second matrix:- The element in the first row, first column: \(2 \times 2 + (-5) \times 0 = 4\).- The element in the first row, second column: \(2 \times (-5) + (-5) \times 7 = -10 - 35 = -45\).- The element in the second row, first column: \(0 \times 2 + 7 \times 0 = 0\).- The element in the second row, second column: \(0 \times (-5) + 7 \times 7 = 49\).Thus, \(A^2\) is:\[A^2 = \left[\begin{array}{rr}{4} & {-45} \ {0} & {49}\end{array}\right]\]

Step 4: Verify the dimensions of the result

After multiplying \(A\) by itself, the resulting matrix has the dimensions \(2 \times 2\), which is consistent with the original matrix \(A\). This confirms that the operation was performed correctly.

Key Concepts

Matrix Algebra

Matrix algebra is a fascinating branch of mathematics that deals with the operations involving matrices. The core operations in matrix algebra include addition, subtraction, and multiplication of matrices. Among these, matrix multiplication is a bit different from regular arithmetic multiplication.
In matrix multiplication, you must align the rows of the first matrix with the columns of the second matrix, ensuring they match in number. This alignment allows the operation to proceed, unlike addition or subtraction, which require matrices of identical dimensions.
For example, let’s consider the multiplication of matrix \( A \) with itself to find \( A^2 \). First, it's crucial to verify if the operation is even possible. Since both matrices involved are \( 2 \times 2 \) in dimensions, multiplication is indeed feasible. After performing the algebraic operation, we end up with another \( 2 \times 2 \) matrix, which underscores the significance of matrix algebra in solving real-world problems involving data structured in rows and columns.

Matrices and Determinants

Matrices are rectangular arrays of numbers organized in rows and columns, playing a foundational role in various mathematical computations. A determinant, on the other hand, is a scalar value that can be computed from the elements of a square matrix. When it comes to matrices, not every array has a determinant.
Determinants are only applicable to square matrices, making them a crucial concept in linear algebra, particularly in solving systems of linear equations, finding inverses, and understanding matrix transformations.

• Determinant of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is given by \( ad-bc \).
• If the determinant is zero, the matrix is said to be singular, indicating no inverse exists.

Understanding both matrices and determinants is essential to grasp advanced mathematical concepts used in various scientific and engineering fields.

Square Matrix

A square matrix is one where the number of rows equals the number of columns. This type of matrix holds special importance in matrix operations, particularly regarding determinants and eigenvalues. A \( 2 \times 2 \) matrix, such as matrix \( A \) from our example, is the simplest form of a square matrix.
In many mathematical processes, square matrices serve as the basis for higher-level concepts like matrix powers, inverses, and determinants:

• Matrix powers involve multiplying a square matrix by itself repeatedly.
• An inverse of a square matrix \( A \) exists only if its determinant is non-zero, solving the equation \( A \times A^{-1} = I \), where \( I \) is the identity matrix.

Square matrices are extensively used in linear transformations, control systems, and computational simulations.

Matrix Dimensions

Matrix dimensions are an essential aspect of matrices, referring to the number of rows and columns they possess. The dimensions are expressed as (rows \( \times \) columns). For instance, a matrix with 2 rows and 3 columns will have dimensions \( 2 \times 3 \).
Understanding matrix dimensions is critical because it determines the rules of matrix operations. In multiplication, for example, the column count of the first matrix must match the row count of the second. This is why in our original problem, calculating \( A^2 \) was straightforward, as both matrices \( A \) being multiplied were \( 2 \times 2 \).

• The product of two matrices will typically result in a matrix with dimensions derived from the outer dimensions of the factors involved.
• Matrix dimensions also dictate compatibility for addition and subtraction, requiring both matrices to have identical dimensions.

By thoroughly understanding matrix dimensions, students can easily predict the feasibility and outcome of matrix operations, ensuring more effective mathematical problem-solving.