Question

Exercises 1-5: For the given matrix functions \(A(t), B(t)\), and \(\mathbf{c}(t)\), make the indicated calculations $$ A(t)=\left[\begin{array}{cc} t-1 & t^{2} \\ 2 & 2 t+1 \end{array}\right], \quad B(t)=\left[\begin{array}{cc} t & -1 \\ 0 & t+2 \end{array}\right], \quad \mathbf{c}(t)=\left[\begin{array}{c} t+1 \\ -1 \end{array}\right] $$ 2 A(t)-3 t B(t)

Short answer

Question: Determine the matrix resulting from the operation \(2A(t)-3tB(t)\), where \(A(t)=\left[\begin{array}{cc} t-1 & t^{2} \\ 2 & 2t+1 \end{array}\right]\) and \(B(t)=\left[\begin{array}{cc} t & -1 \\ 0 & t+2 \end{array}\right]\). Answer: The resulting matrix is \(\left[\begin{array}{cc} 3t^{2}+2t-2 & 2t^{2}-3t \\ 4 & 3t^{2}+10t+2 \end{array}\right]\).

Step-by-step solution

Recall matrix addition and scalar multiplication rules

When adding or subtracting matrices, elements with the same position are added or subtracted. When multiplying a matrix by a scalar, each element of the matrix is multiplied by the scalar. In this exercise we have to multiply \(A(t)\) by 2, \(B(t)\) by \(-3t\), and then subtract the resulting matrices.

Multiply A(t) by 2

Multiply each element of \(A(t)\) by 2: $$ 2A(t) = 2\left[\begin{array}{cc} t-1 & t^{2} \\ 2 & 2t+1 \end{array}\right] = \left[\begin{array}{cc} 2(t-1) & 2t^{2} \\ 4 & 4t+2 \end{array}\right] $$

Multiply B(t) by -3t

Multiply each element of \(B(t)\) by \(-3t\): $$ -3tB(t) = -3t\left[\begin{array}{cc} t & -1 \\ 0 & t+2 \end{array}\right] = \left[\begin{array}{cc} -3t^{2} & 3t \\ 0 & -3t(t+2) \end{array}\right] $$

Subtract the matrices

Subtract each element of the resulting \(-3tB(t)\) from the corresponding element of \(2A(t)\): $$ 2A(t) - 3tB(t) = \left[\begin{array}{cc} 2(t-1) & 2t^{2} \\ 4 & 4t+2 \end{array}\right] - \left[\begin{array}{cc} -3t^{2} & 3t \\ 0 & -3t(t+2) \end{array}\right] = \left[\begin{array}{cc} 2(t-1)+3t^{2} & 2t^{2}-3t \\ 4 & 4t+2+3t(t+2) \end{array}\right] $$

Simplify the result

Perform arithmetic operations to simplify the resulting matrix: $$ 2A(t) - 3tB(t) = \left[\begin{array}{cc} 2t-2+3t^{2} & -3t+2t^{2} \\ 4 & 4t+2+3t^{2}+6t \end{array}\right] = \left[\begin{array}{cc} 3t^{2}+2t-2 & 2t^{2}-3t \\ 4 & 3t^{2}+10t+2 \end{array}\right] $$

Key concepts

Matrix Operations

Matrix operations are fundamental tools in linear algebra used to manipulate and transform matrices. They include operations like addition, subtraction, multiplication (including scalar multiplication), and finding determinants. When you perform matrix operations, it's important to follow certain rules to ensure correctness:
- **Addition and Subtraction**: These operations require matrices to be of the same dimension. You add or subtract corresponding elements from each matrix.
- **Multiplication**: Unlike addition and subtraction, multiplication between two matrices requires the number of columns in the first matrix to match the number of rows in the second matrix.
Performing these operations correctly is crucial for more complex matrix equations.

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a single number, called a scalar. This operation is straightforward and involves scaling the whole matrix by the scalar. Let's look at how this applies practically:
- **Example**: If you have a matrix \(\begin{bmatrix}a & b \c & d\end{bmatrix}\) and you multiply it by a scalar \(k\), each element becomes: \(\begin{bmatrix}ka & kb \kc & kd\end{bmatrix}\)- **Applications**: This technique is useful for balancing equations, modeling scenarios where each unit impacts equally, or adjusting scales when aligning datasets.

Matrix Subtraction

Matrix subtraction is another fundamental operation where two matrices of the same dimensions are subtracted element-wise. This means each element in one matrix is subtracted from the corresponding element in the other matrix.
Let's break it down:
- To perform matrix subtraction, ensure both matrices have the same size. If not, subtraction isn’t possible.
- Similar to scalar multiplication, line up each element and perform the subtraction: \[\begin{bmatrix}a & b \c & d\end{bmatrix}- \begin{bmatrix}e & f \g & h\end{bmatrix}= \begin{bmatrix}a-e & b-f \c-g & d-h\end{bmatrix}\]This operation is particularly useful when you need to compare datasets or solve systems of equations.

Matrix Simplification

Matrix simplification is the process of making a resulting matrix expression easier to interpret and use. This often comes after multiple operations like addition, subtraction, or multiplication.
- **Simplifying Expressions**: After performing operations, you combine like terms and perform arithmetic calculations to reduce each entry to its simplest form. For example, combining terms or constants,
- **Reducing Complexity**: This not only makes the matrix more manageable but also makes patterns and solutions more visible. - **Applications**: Simplification is essential in predictive modeling, econometrics, or anytime data needs to be interpreted quickly and accurately.

By practicing simplification techniques, you'll be able to deconstruct complex matrices and use them for effective decision-making.