Problem 34 - Precalculus

Question

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1 \end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2 \end{array}\right]$$ $$E=\left[\begin{array}{rrr}-1 & 0 & 1 \\\2 & 1 & 4 \\\\-3 & 1 & 5 \end{array}\right] \quad F=\left[\begin{array}{r}1 \\\0 \\\\-1\end{array}\right] \quad G=\left[\begin{array}{ll}1 & 2 \\\3 & 4\end{array}\right]$$ $$-3 E D$$

Step-by-Step Solution

Answer

The result of \(-3ED\) is \(\begin{bmatrix} -6 & -3 \\ -60 & 39 \\ -42 & 0 \end{bmatrix}\).

Step 1: Determine Dimension Compatibility

First, we need to verify if the operation \(-3ED\) is possible by checking the dimensions of matrices \(E\) and \(D\). Matrix \(E\) is a \(3 \times 3\) matrix, and matrix \(D\) is a \(3 \times 2\) matrix. Since the number of columns in \(E\) (which is 3) matches the number of rows in \(D\) (also 3), the multiplication \(ED\) is feasible. The resulting matrix will have dimensions \(3 \times 2\). Multiplying by a scalar \(-3\) after the matrix multiplication is always valid.

Step 2: Perform Matrix Multiplication ED

Calculate the matrix product \(ED\) by multiplying each row of matrix \(E\) with each column of matrix \(D\):Let \(E = \begin{bmatrix} -1 & 0 & 1 \ 2 & 1 & 4 \ -3 & 1 & 5 \end{bmatrix}\) and \(D = \begin{bmatrix} 2 & -3 \ 0 & 1 \ 4 & -2 \end{bmatrix}\).Calculate each element of the resulting \(3 \times 2\) matrix:1. For the first row: - First column: \((-1)(2) + (0)(0) + (1)(4) = -2 + 4 = 2\) - Second column: \((-1)(-3) + (0)(1) + (1)(-2) = 3 - 2 = 1\)2. For the second row: - First column: \((2)(2) + (1)(0) + (4)(4) = 4 + 16 = 20\) - Second column: \((2)(-3) + (1)(1) + (4)(-2) = -6 + 1 - 8 = -13\)3. For the third row: - First column: \((-3)(2) + (1)(0) + (5)(4) = -6 + 20 = 14\) - Second column: \((-3)(-3) + (1)(1) + (5)(-2) = 9 + 1 - 10 = 0\)Thus, \(ED = \begin{bmatrix} 2 & 1 \ 20 & -13 \ 14 & 0 \end{bmatrix}\).

Step 3: Multiply the Result by -3

Now, multiply the resulting matrix by the scalar \(-3\):\[-3 \begin{bmatrix} 2 & 1 \ 20 & -13 \ 14 & 0 \end{bmatrix} = \begin{bmatrix} -3 \times 2 & -3 \times 1 \ -3 \times 20 & -3 \times -13 \ -3 \times 14 & -3 \times 0 \end{bmatrix} = \begin{bmatrix} -6 & -3 \ -60 & 39 \ -42 & 0 \end{bmatrix}\].

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra where two matrices are multiplied together to produce a new 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. For example, when multiplying a \(3 \times 3\) matrix with a \(3 \times 2\) matrix, the inner dimensions (both "3") match, allowing the operation to proceed. The result is a \(3 \times 2\) matrix.

The product of matrix multiplication is found by taking each element of a row in the first matrix and multiplying it with each element of a column in the second matrix, then summing these products. This process is repeated for each row of the first matrix and each column of the second matrix.

• To multiply row by column: compute elements like \((-1)(2) + (0)(0) + (1)(4) = 2\).
• Repeat for all columns and rows to complete the new matrix.

Understanding matrix multiplication helps solve complex problems in various fields such as computer graphics and engineering.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a single real number, known as the scalar. This operation is straightforward and involves a simple arithmetic calculation for each element.

Imagine you have a resulting matrix from a multiplication operation like \(ED\), and you then multiply this matrix by a scalar \(-3\). For each element in the matrix, you would perform a calculation such as \(-3 \times 2 = -6\).

• This operation affects all elements equally, scaling the entire matrix to make it larger or smaller numerically.
• For each entry in the result matrix, apply the scalar transformation individually.

This operation is useful when you need to adjust the scale of a matrix without changing its form. It has practical applications in physics and other sciences.

Matrix Dimensions

Matrix dimensions are a description of the size of a matrix in terms of rows and columns. A matrix with dimensions \(m \times n\) has \(m\) rows and \(n\) columns. Knowing these dimensions is crucial for successful matrix operations, particularly when verifying that a multiplication operation is possible.

In the given context, for the matrices \(E\) and \(D\):

• Matrix \(E\) is a \(3 \times 3\) matrix, which means it has 3 rows and 3 columns.
• Matrix \(D\) is a \(3 \times 2\) matrix, adorned with 3 rows and 2 columns.

If you want to multiply these two, check first whether the number of columns in \(E\) matches the number of rows in \(D\). This matching is what allows the matrix multiplication to proceed, resulting in a new matrix of dimensions \(3 \times 2\). Understanding dimensions is key to performing correct matrix operations and helps prevent illegal operations, such as multiplying non-compatible matrices.

Linear Algebra

Linear algebra is a branch of mathematics concerned with vector spaces and linear mappings between them. It includes matrix operations as one of its fundamental topics. Linear algebra provides frameworks for understanding how to perform and understand transformations, solve systems of linear equations, and work with multi-dimensional data.

Core principles include:

• Vectors are often represented as rows or columns in matrices.
• Matrix multiplication, addition, and scalar multiplication are primary operations that help denote these transformations.
• Applications of linear algebra extend to machine learning, quantum mechanics, and much more.

In the problem we've approached with multiplying matrices and scalars, we're applying these principles to understand how to manipulate and transform data efficiently. Gaining a solid understanding of linear algebra helps in diverse scientific and engineering fields, providing tools to tackle complex problems involving multi-dimensional data.