Problem 18 - Precalculus

Question

Use the matrices \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] \quad B=\left[\begin{array}{rr}0 & -3 \\ -5 & 2\end{array}\right] \quad C=\left[\begin{array}{rrr}10 & -\frac{11}{2} & 0 \\ \frac{3}{5} & 5 & 9\end{array}\right]\) \(D=\left[\begin{array}{rr}7 & -13 \\ -\frac{4}{3} & 0 \\\ 6 & 8\end{array}\right] \quad E=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & 4 & -9 \\ 0 & 0 & -5\end{array}\right]\) to compute the following or state that the indicated operation is undefined. $$ E^{2}+5 E-36 I_{3} $$

Step-by-Step Solution

Answer

The result is the matrix \( \begin{bmatrix}-30 & 20 & -6 \\ 0 & 0 & -126 \\ 0 & 0 & -36\end{bmatrix} \).

Step 1: Understand the Expression

We need to compute \(E^2 + 5E - 36I_3\), where \(E\) is a \(3 \times 3\) matrix and \(I_3\) is the \(3 \times 3\) identity matrix. This involves calculating the square of matrix \(E\), scaling it and matrices \(E\) and \(I_3\), and then performing matrix addition.

Step 2: Compute E^2

To compute \(E^2\), multiply matrix \(E\) by itself:\[E \times E = \begin{bmatrix}1 & 2 & 3 \ 0 & 4 & -9 \ 0 & 0 & -5\end{bmatrix} \times \begin{bmatrix}1 & 2 & 3 \ 0 & 4 & -9 \ 0 & 0 & -5\end{bmatrix}\]The result is:\[E^2 = \begin{bmatrix}1 & 2 & 3 \ 0 & 4 & -9 \ 0 & 0 & -5\end{bmatrix} \times \begin{bmatrix}1 & 2 & 3 \ 0 & 4 & -9 \ 0 & 0 & -5\end{bmatrix} = \begin{bmatrix}1(1) + 2(0) + 3(0) & 1(2) + 2(4) + 3(0) & 1(3) + 2(-9) + 3(-5) \ 0 & 16 & -36-45 \ 0 & 0 & 25\end{bmatrix} = \begin{bmatrix}1 & 10 & -21 \ 0 & 16 & -81 \ 0 & 0 & 25\end{bmatrix}\]

Step 3: Compute 5E

Scale the matrix \(E\) by 5:\[5E = 5 \times \begin{bmatrix}1 & 2 & 3 \ 0 & 4 & -9 \ 0 & 0 & -5\end{bmatrix} = \begin{bmatrix}5 & 10 & 15 \ 0 & 20 & -45 \ 0 & 0 & -25\end{bmatrix}\]

Step 4: Compute 36I_3

Scale the identity matrix \(I_3\) by 36:\[36I_3 = 36 \times \begin{bmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}36 & 0 & 0 \ 0 & 36 & 0 \ 0 & 0 & 36\end{bmatrix}\]

Step 5: Add the Matrices

Add the matrices \(E^2\), \(5E\), and \(-36I_3\):\[E^2 + 5E - 36I_3 = \begin{bmatrix}1 & 10 & -21 \ 0 & 16 & -81 \ 0 & 0 & 25\end{bmatrix} + \begin{bmatrix}5 & 10 & 15 \ 0 & 20 & -45 \ 0 & 0 & -25\end{bmatrix} - \begin{bmatrix}36 & 0 & 0 \ 0 & 36 & 0 \ 0 & 0 & 36\end{bmatrix}\]Combine the elements:\[= \begin{bmatrix}1 + 5 - 36 & 10 + 10 - 0 & -21 + 15 - 0 \ 0 + 0 - 0 & 16 + 20 - 36 & -81 - 45 - 0 \ 0 + 0 - 0 & 0 + 0 - 0 & 25 - 25 - 36\end{bmatrix}\]\[= \begin{bmatrix}-30 & 20 & -6 \ 0 & 0 & -126 \ 0 & 0 & -36\end{bmatrix}\]

Key Concepts

Matrix Multiplication

Matrix multiplication is one of the cornerstone operations in linear algebra. It involves taking two matrices and producing a new matrix. However, for this to be possible, specific conditions must be met. Primarily, for matrices A and B, if A is an m × n matrix, B must be an n × p matrix. This means the number of columns in the first matrix must match the number of rows in the second. The resulting matrix will be an m × p matrix.

To perform matrix multiplication, each element in the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and column from the second. The equation is:
- Multiply corresponding entries and then sum them up for each position.- Example: For matrices A = \([a_{ij}]\) and B = \([b_{ij}]\), the element \(c_{ij}\) in their product C is given by \(c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \ldots + a_{in}b_{nj}\) where the sum is taken over all columns of A and rows of B.Fixed properly, multiplication is widely used for transforming data and finding relationships in structures.

Matrix Addition

Matrix addition is a more straightforward operation compared to multiplication. It involves adding corresponding elements of matrices that are of the same size to produce another matrix of the same size. If you have two matrices of dimensions m × n, their sum will also be an m × n matrix.

Each element in the resulting matrix is simply the sum of the corresponding elements from the matrices being added:

• Align matrices by their dimensions.
• Add elements in the same position directly.

For example, for matrices A = \([a_{ij}]\) and B = \([b_{ij}]\), their sum C = \([c_{ij}]\) is computed as \(c_{ij} = a_{ij} + b_{ij}\) for each element.
This operation is visually analogous to creating a new image by layering two identically sized transparencies on top of one another.

Identity Matrix

The identity matrix is a special kind of square matrix that plays a crucial role in matrix operations, especially in maintaining the integrity of data during transformations. An identity matrix is a square matrix in which all the elements of the principal diagonal are ones, and all other elements are zeros. It is denoted as I.

The identity matrix holds a pivotal property:

• When a matrix is multiplied by the identity matrix, its value remains unchanged, much like multiplying a number by one.

For a matrix A, the equation \(A imes I = I imes A = A\) holds true, where I is the identity matrix of suitable dimensions. Identifying and utilizing the identity matrix is fundamental in solving equations and simplifying expressions in linear algebra.

Matrix Scaling

Matrix scaling is the process of multiplying each element within a matrix by a scalar (a single number). This operation modifies each entry of the matrix but keeps its structure (the arrangement of rows and columns) intact.

Given a matrix A and a scalar k, the scaled matrix is formed by multiplying each element of A by k. Mathematically, if matrix A has elements \(a_{ij}\), the scaled matrix will have elements \(k imes a_{ij}\). This operation can be visualized similarly to adjusting the brightness of an entire image.

The application of scaling is vast, ranging from solving linear equations to manipulating geometry in graphics. It's important to understand how scaling affects matrices, enabling greater control and accuracy in various mathematical problems.