Problem 29 - Precalculus with Limits

Question

In Exercises 29-32, solve for \(X\) in the equation, given \(A=\left[\begin{array}{r} -2 & -1 \\ 1 & 0 \\ 3 & -4 \end{array}\right]\) and \(B=\left[\begin{array}{r} 0 & 3 \\ 2 & 0 \\ -4 & -1 \end{array}\right]\) \(X = 3A - 2B\)

Step-by-Step Solution

Answer

The matrix \(X\) is \[ \left[\begin{array}{r} -6 & -12 \ 1 & 6 \ 18 & -5 \end{array}\right] \]

Step 1: Scaling of matrix A

Firstly multiply every element of matrix \(A\) by 3. This will produce a new matrix, call it \(A'\), in which each of \(A\)'s elements have been multiplied by 3.

Step 2: Scaling of matrix B

Now, multiply every element of matrix \(B\) by -2. This will produce a new matrix, call it \(B'\), in which each of \(B\)'s elements have been multiplied by -2.

Step 3: Subtract matrix B' from matrix A'

Subtract each element of matrix \(B'\) from the corresponding element in matrix \(A'\). The resulting matrix is \(X\). Each element in matrix \(X\) is thus the result of the operation \(A'_ij - B'_ij\) where \(i\) denotes the row number, and \(j\) denotes the column number of the element in question.

Key Concepts

Matrix Subtraction

Matrix subtraction is a crucial concept in matrix algebra, where you subtract corresponding elements of one matrix from another. For two matrices to be subtractable, they must have the same dimensions. This means that both matrices must have the same number of rows and columns.
Consider two matrices, say matrix C and matrix D. If C and D are both 2x3 matrices, the subtraction is straightforward:

• Subtract the first element of matrix D from the first element of matrix C.
• Continue element by element, moving across rows and columns.

Written mathematically:\[X_{ij} = C_{ij} - D_{ij}\] Here, \(X_{ij}\) denotes the element in the resulting matrix X after subtraction, and it is located in the same position \(i, j\) as the elements \(C_{ij}\) in matrix C and \(D_{ij}\) in matrix D. It's imperative to note that matrix subtraction is performed element-wise, and handling matrices of different dimensions will result in an error.

Matrix Scaling

Matrix scaling, also known as scalar multiplication of a matrix, involves multiplying a matrix by a scalar (a constant number). This operation changes the magnitude of the matrix without altering its dimensions or structure. To perform scalar multiplication, you multiply each element of the matrix by the given scalar.
Take, for instance, a matrix E and a scalar k. The scaled matrix, let's call it F, is obtained by multiplying each element of matrix E by k:

• Multiply each element of the first row of matrix E by the scalar \(k\).
• Repeat the process for each subsequent row.

This scaling is expressed mathematically as:\[F_{ij} = k imes E_{ij}\] Here, \(F_{ij}\) represents the element at position \(i, j\) in the resulting matrix F, which is obtained by multiplying \(E_{ij}\), the element in matrix E at the same position, by the scalar k. Through scaling, you can effectively increase or decrease the intensity or size represented by the matrix without changing its layout.

Linear Transformations

Linear transformations are functions from one vector space to another that preserve the operations of vector addition and scalar multiplication. In the context of matrices, they map input vectors to output vectors, transforming them according to the characteristics of the matrix.
For a matrix M that represents a linear transformation, applying M to a vector v involves matrix multiplication, giving a new vector w:\[w = Mv\]This equation demonstrates how a vector v is transformed into a new vector w by the matrix M.

• Linear transformations can include operations like rotations, reflections, scaling, and translations in a plane or space.
• They keep grid lines parallel and evenly spaced.

Linear transformations are represented uniquely by matrices due to their structural properties. To prove a transformation is linear, it must fulfill two conditions:

• Additivity: \(T(u + v) = T(u) + T(v)\) for any vectors u and v.
• Homogeneity: \(T(cv) = cT(v)\) for any vector v and scalar c.

These conditions ensure the transformation respects the addition and scalar multiplication inherent to the vector space's structure. Understanding linear transformations is key to grasping concepts in higher mathematics and applications such as computer graphics and systems of equations.