Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-1 & -2 \\\5 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 7.

Step 1: Identify the Matrix

We are given a 2x2 matrix: \[ \begin{bmatrix} -1 & -2 \ 5 & 3 \end{bmatrix} \]. The matrix is of size 2x2, which means it has 2 rows and 2 columns.

Step 2: Use the Determinant Formula for a 2x2 Matrix

The formula for finding the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by: \[ \text{det} = ad - bc \]. For our matrix, this translates to \( a = -1 \), \( b = -2 \), \( c = 5 \), and \( d = 3 \).

Step 3: Perform the Multiplications

Using the formula, calculate the products \( ad \) and \( bc \): - Calculate \( ad = (-1) \times 3 = -3 \).- Calculate \( bc = (-2) \times 5 = -10 \).

Step 4: Substitute and Solve

Substitute the products back into the determinant formula: \[ \text{det} = (-3) - (-10) \]. Since subtracting a negative is the same as adding, this simplifies to: \[ \text{det} = -3 + 10 = 7 \].

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple structure in mathematics used to organize numbers, variables, or expressions in a grid with two rows and two columns. It's often represented as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), where \(a\), \(b\), \(c\), and \(d\) are elements that can be any real or complex number.
Understanding a 2x2 matrix is fundamental in linear algebra because:

• It forms the basis for more complex matrices of larger dimensions.
• Matrices are useful in various applications such as solving systems of linear equations.

In our case, the matrix \( \begin{bmatrix} -1 & -2 \ 5 & 3 \end{bmatrix} \) clearly shows its elements in a neatly organized manner, making further computations straightforward.

Matrix Multiplication

Matrix multiplication is an operation where two matrices are multiplied to produce a third matrix. It's important in many applications in linear algebra, computer graphics, and more. However, the determinant solving featured here is not part of regular matrix multiplication, but understanding multiplication is crucial when dealing with matrix operations.
When multiplying matrices, the number of columns in the first matrix must match the number of rows in the second matrix. With 2x2 matrices, this often involves multiplying corresponding elements and summing them up correctly. For instance:

• If multiplying: \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) with \( \begin{bmatrix} e & f \ g & h \end{bmatrix} \), you perform these operations: \( ae+bg \), \( af+bh \), \( ce+dg \), and \( cf+dh \).

It reinforces how familiarizing oneself with elements' positions is crucial in matrix operations.

Linear Algebra

Linear algebra is a branch of mathematics that explores vectors, vector spaces, and linear transformations among others. It's the framework for solving systems of linear equations. Matrices are a central component of linear algebra, serving as representations of linear transformations or data organization tools.
Matrix operations, such as calculating determinants, are essential parts of linear algebra since they provide insights into properties such as invertibility and solutions to equations.

• Determinants help assess if a matrix has an inverse. A non-zero determinant indicates an invertible matrix, which is critical in solving equations efficiently.
• In computer graphics, matrices and their determinants are used for transformations like rotation and scaling.

In our exercise, understanding how to calculate the determinant of a 2x2 matrix strengthens the foundation for more advanced linear algebra concepts.

Matrix Determinant Formula

The determinant of a matrix is a special number that gives insights into the matrix's properties, such as its invertibility or the volume it transforms. Specifically, a 2x2 matrix determinant helps in understanding these characteristics effortlessly.
The formula for a 2x2 matrix, provided as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), is \( \text{det} = ad - bc \). This directly relates the matrix entries in a combination that reflects its inherent transformation properties. For the matrix \( \begin{bmatrix} -1 & -2 \ 5 & 3 \end{bmatrix} \):

• First, multiply the diagonal elements, \( (-1) \times 3 = -3 \).
• Then, multiply the off-diagonal elements, \( (-2) \times 5 = -10 \).
• Subtract these products to find the determinant: \( -3 - (-10) = 7 \).

Understanding and applying this formula is crucial in linear algebra tasks, providing a quick test of invertibility and connecting to deeper algebraic applications.