Question

Compute the following: a. \(\left[\begin{array}{lll}3 & 2 & 1 \\ 5 & 1 & 0\end{array}\right]-5\left[\begin{array}{rrr}3 & 0 & -2 \\ 1 & -1 & 2\end{array}\right]\) b. \(3\left[\begin{array}{r}3 \\ -1\end{array}\right]-5\left[\begin{array}{l}6 \\\ 2\end{array}\right]+7\left[\begin{array}{r}1 \\ -1\end{array}\right]\) c. \(\left[\begin{array}{rr}-2 & 1 \\ 3 & 2\end{array}\right]-4\left[\begin{array}{rr}1 & -2 \\ 0 & -1\end{array}\right]+3\left[\begin{array}{rr}2 & -3 \\ -1 & -2\end{array}\right]\) d. \(\left[\begin{array}{lll}3 & -1 & 2\end{array}\right]-2\left[\begin{array}{lll}9 & 3 & 4\end{array}\right]+\left[\begin{array}{lll}3 & 11 & -6\end{array}\right]\) e. \(\left[\begin{array}{rrrr}1 & -5 & 4 & 0 \\ 2 & 1 & 0 & 6\end{array}\right]^{T}\) f. \(\left[\begin{array}{rrr}0 & -1 & 2 \\ 1 & 0 & -4 \\ -2 & 4 & 0\end{array}\right]^{T}\) g. \(\left[\begin{array}{rr}3 & -1 \\ 2 & 1\end{array}\right]-2\left[\begin{array}{rr}1 & -2 \\ 1 & 1\end{array}\right]^{T}\) h. \(3\left[\begin{array}{rr}2 & 1 \\ -1 & 0\end{array}\right]^{T}-2\left[\begin{array}{rr}1 & -1 \\ 2 & 3\end{array}\right]\)

Short answer

Solutions: (a) \([-12, 2, 11; 0, 6, -10]\), (b) \([-14; -20]\), (c) \([0, 0; 6, 0]\), (d) \([-12, 4, -12]\), (e) Transpose: \([1, 2; -5, 1; 4, 0; 0, 6]\), (f) Transpose: \([0, 1, -2; -1, 0, 4; 2, -4, 0]\), (g) \([1, -3; 6, -1]\), (h) \([4, -1; -1, -6]\).

Step-by-step solution

Exercise Breakdown

The problem involves multiple computations with matrices. We'll approach each part (a-h) separately, following the instructions related to matrix operations such as scalar multiplication, addition, transposition, etc.

Solve Part (a)

Subtract the scaled matrix from the first matrix.\[\left[\begin{array}{lll}3 & 2 & 1 \ 5 & 1 & 0\end{array}\right]-5\left[\begin{array}{rrr}3 & 0 & -2 \ 1 & -1 & 2\end{array}\right]=\left[\begin{array}{rrr}3-15 & 2 & 1+10 \ 5-5 & 1+5 & 0-10\end{array}\right]=\left[\begin{array}{rrr}-12 & 2 & 11 \ 0 & 6 & -10\end{array}\right]\]

Solve Part (b)

Calculate the result of linear combinations of column vectors.\[3\left[\begin{array}{r}3 \ -1\end{array}\right]-5\left[\begin{array}{l}6 \ 2\end{array}\right]+7\left[\begin{array}{r}1 \ -1\end{array}\right]=\left[\begin{array}{r}9 \ -3\end{array}\right]-\left[\begin{array}{l}30 \ 10\end{array}\right]+\left[\begin{array}{r}7 \ -7\end{array}\right]=\left[\begin{array}{r}-14 \ -20\end{array}\right]\]

Solve Part (c)

Compute the matrix expression using distribution and addition.\[\left[\begin{array}{rr}-2 & 1 \ 3 & 2\end{array}\right]-4\left[\begin{array}{rr}1 & -2 \ 0 & -1\end{array}\right]+3\left[\begin{array}{rr}2 & -3 \ -1 & -2\end{array}\right]=\left[\begin{array}{rr}-2-4+6 & 1+8-9 \ 3+0+3 & 2+4-6\end{array}\right]=\left[\begin{array}{rr}0 & 0 \ 6 & 0\end{array}\right]\]

Solve Part (d)

Perform the scalar multiplication and addition operations.\[\left[\begin{array}{lll}3 & -1 & 2\end{array}\right]-2\left[\begin{array}{lll}9 & 3 & 4\end{array}\right]+\left[\begin{array}{lll}3 & 11 & -6\end{array}\right]=\left[\begin{array}{lll}3-18+3 & -1-6+11 & 2-8-6\end{array}\right]=\left[\begin{array}{lll}-12 & 4 & -12\end{array}\right]\]

Solve Part (e)

Transpose the given matrix.\[\left[\begin{array}{rrrr}1 & -5 & 4 & 0 \ 2 & 1 & 0 & 6\end{array}\right]^{T}=\left[\begin{array}{rr}1 & 2 \ -5 & 1 \ 4 & 0 \ 0 & 6\end{array}\right]\]

Solve Part (f)

Transpose the matrix.\[\left[\begin{array}{rrr}0 & -1 & 2 \ 1 & 0 & -4 \ -2 & 4 & 0\end{array}\right]^{T}=\left[\begin{array}{rrr}0 & 1 & -2 \ -1 & 0 & 4 \ 2 & -4 & 0\end{array}\right]\]

Solve Part (g)

Subtract the transposed and scaled matrix from the original matrix.\[\left[\begin{array}{rr}3 & -1 \ 2 & 1\end{array}\right]-2\left[\begin{array}{rr}1 & -2 \ 1 & 1\end{array}\right]^{T}=\left[\begin{array}{rr}3 & -1 \ 2 & 1\end{array}\right]-\left[\begin{array}{rr}2 & 2 \ -4 & 2\end{array}\right]=\left[\begin{array}{rr}1 & -3 \ 6 & -1\end{array}\right]\]

Solve Part (h)

Compute the matrix subtraction and transposition.\[3\left[\begin{array}{rr}2 & 1 \ -1 & 0\end{array}\right]^{T}-2\left[\begin{array}{rr}1 & -1 \ 2 & 3\end{array}\right]=\left[\begin{array}{rr}6 & -3 \ 3 & 0\end{array}\right]-\left[\begin{array}{rr}2 & -2 \ 4 & 6\end{array}\right]=\left[\begin{array}{rr}4 & -1 \ -1 & -6\end{array}\right]\]

Key concepts

Matrix Operations

Matrix operations are fundamental procedures in linear algebra, allowing various manipulations of matrices to solve equations or model scenarios. These operations include adding, subtracting, multiplying matrices by each other or by scalars, and transposing matrices. Understanding these processes is essential for grasping higher-level mathematics and applications in many fields such as physics, engineering, and computer science.

Matrix operations perform arithmetic on matrices:

• Addition and subtraction require matrices of the same dimensions.
• Multiplication involves combining two matrices, or one matrix may be multiplied by a scalar (a single number), which we call scalar multiplication.
• Transposition changes the orientation of a matrix, flipping it over its diagonal.

In problems and real-world applications, combining these operations helps in solving complex systems or converting data. Mastery of these tasks gives you a toolkit for evaluating and transforming matrices to derive meaningful results from a set of data.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar (single constant value). This is akin to simple multiplication but applied across the entire set of a matrix's entries. Given a matrix \(A\) = \(\begin{bmatrix}a & b \c & d \end{bmatrix}\), multiplying it by a scalar \(k\), the resulting matrix is \(kA = \begin{bmatrix}ka & kb \kc & kd \end{bmatrix}\).

This process alters the size or magnitude of the matrix but not its shape. It is widely used in linear transformations and is beneficial in real-world applications where scaling of data is required. When dealing with matrix operations in exercises, scalar multiplication often comes before addition or subtraction to change the scale of one or more matrices involved in the operation. By doing so, it ensures matrices align correctly for subsequent operations.

Transpose of a Matrix

Transposing a matrix is simple: it means flipping a matrix over its diagonal. The columns of the original matrix become the rows of the new one, and vice versa. For a given matrix \(A = \begin{bmatrix} a & b & c \ d & e & f \end{bmatrix}\), the transpose, written as \(A^T\), is \(\begin{bmatrix} a & d \ b & e \ c & f \end{bmatrix}\).

Transposing is crucial for certain mathematical operations and helps in solving equations where differential dimensions require alignment. It is like rewriting the matrix from another perspective, and it is particularly useful in dot products and cross products. Whenever performing operations like matrix multiplication, transposition helps maintain compatibility of structures, especially in vector space calculations.

• Transposing flips "width into height" and "height into width".
• This operation is also an important precursor to determining matrices' inverses and adjoints.

A well-grasped transposition concept streamlines many other complex matrix operations.

Matrix Addition and Subtraction

Matrix addition and subtraction are performed similarly to addition and subtraction of regular numbers, except each element in the corresponding position must be of the same matrix dimension. Two matrices can be added or subtracted only if they have the same dimensions. The process involves adding (or subtracting) the elements with matching positions in each matrix.

Consider two matrices, \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\). The sum and difference are \(A + B = \begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix}\), and \(A - B = \begin{bmatrix} a-e & b-f \ c-g & d-h \end{bmatrix}\).

This operation is vital because it allows combining matrices symbolizing cumulative effects or differences (such as finding net forces when working with physics vectors).

• Remember: Sum and difference calculations are element-wise.
• Ensure matrix dimensions align before performing these operations.

Mastering these, along with other matrix operations, provides a robust foundation for further studying complex systems and simulations.