Problem 35 - 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. $$B^{2}$$

Step-by-Step Solution

Answer

The operation \( B^2 \) cannot be performed because \( B \) is a 2x3 matrix, and its dimensions are incompatible for squaring.

Step 1: Understand the Operation

The operation given is to find the square of matrix \( B \), denoted by \( B^2 \). This means multiplying matrix \( B \) by itself.

Step 2: Verify Dimensions for Multiplication

Matrix \( B \) is of size 2x3. For matrix multiplication \( B \times B \) to be possible, the number of columns of the first matrix must equal the number of rows of the second matrix. Since both instances of \( B \) are 2x3, the multiplication is not possible.

Step 3: State Conclusion

The operation \( B^2 \) cannot be performed because matrix \( B \) does not have the required square size (same number of rows and columns) for self-multiplication.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves taking two matrices and producing another matrix. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. This requirement ensures that the multiplication is feasible and can proceed smoothly.

Here’s how matrix multiplication works:

• Take elements from a row of the first matrix.
• Multiply these elements by corresponding elements from a column of the second matrix.
• Sum these products to get an element of the resulting matrix.

Each element in the resulting matrix is derived from this process. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. This is why the condition of matching columns and rows is crucial.

For example, if matrix A is of size 2x3 and matrix B is of size 3x2, you can multiply them because the inner dimensions (3 and 3) match.

Matrices

Matrices are rectangular arrangements of numbers or functions organized into rows and columns. They are used across different fields such as computer science, physics, and economics for various applications, including modeling and transformations.

Matrices can have different sizes depending on how many rows and columns they contain:

• A matrix with only one row is called a row matrix.
• A matrix with only one column is called a column matrix.
• If the number of rows equals the number of columns, it is a square matrix.

Square matrices are particularly significant because they can have special properties, such as determinants and eigenvalues, which are not typically applicable to non-square matrices.

Matrices are denoted by uppercase letters, like matrix A or matrix B, and operations on matrices, such as addition and multiplication, must respect their sizes. For example, you can only add matrices if they have the same dimensions.

Algebraic Operations

Algebraic operations with matrices include addition, subtraction, and multiplication. These operations are similar to numerical operations but have specific rules and restrictions when applied to matrices.

Let's look at some common matrix operations:

• Addition and Subtraction: Only possible when matrices are of the same size. Corresponding elements are added or subtracted together.
• Scalar Multiplication: Every element of a matrix is multiplied by a number (a scalar). Dimensions of the matrix remain unchanged.
• Matrix Multiplication: As discussed earlier, requires matching columns of the first matrix to rows of the second.

Understanding these operations is crucial for solving problems in algebra that involve matrices. Ensuring operations are legitimate based on matrix size and structure is key to successful calculations.