Question

Find the determinant of the \(2 \times 2\) matrix. \(\left[\begin{array}{cc}-1 & -7 \\ -5 & 9\end{array}\right]\)

Short answer

The determinant is -44.

Step-by-step solution

Understand the Matrix Format

The matrix given is of size \(2 \times 2\), which means it has two rows and two columns. The matrix is written as:\[\begin{bmatrix}-1 & -7 \-5 & 9\end{bmatrix}\]

Recall the Determinant Formula for 2×2 Matrices

To find the determinant of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), use the formula:\[\text{det}(A) = ad - bc\]

Identify Matrix Elements

Identify the elements of the matrix:- \(a = -1\)- \(b = -7\)- \(c = -5\)- \(d = 9\)These values will be used in the formula to calculate the determinant.

Substitute the Elements into the Formula

Substitute the identified values into the determinant formula:\[\text{det}(A) = (-1)(9) - (-7)(-5)\]

Calculate the Determinant

Perform the calculations:- First part: \((-1) \times 9 = -9\)- Second part: \((-7) \times (-5) = 35\)Now subtract the second result from the first:\(-9 - 35 = -44\)

Conclude with the Determinant Value

The determinant of the matrix is \(-44\).

Key concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a simple rectangular arrangement with two rows and two columns. It is expressed in the form \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], where each element is part of the matrix's structure. The layout is as follows:

• The top-left element is \( a \).
• The top-right element is \( b \).
• The bottom-left element is \( c \).
• The bottom-right element is \( d \).

Understanding this setup is crucial because it lays the foundation for further calculations, such as finding the determinant. A matrix like this is frequently used in algebra to perform operations such as transformations and system-solving. These matrices are prevalent in fields like computer graphics and engineering.

Determinant Formula for a 2x2 Matrix

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the determinant helps us understand matrix properties such as invertibility. The formula for finding the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by: \[ \text{det}(A) = ad - bc \] This simple yet powerful formula takes the product of the diagonal elements \( a \) and \( d \) and subtracts from that the product of the other diagonal elements \( b \) and \( c \). Using this formula allows us to assess whether a matrix is invertible (a non-zero determinant) or not (a zero determinant). A matrix with a zero determinant is singular and does not have an inverse.

Matrix Elements Identification

Before applying the determinant formula, it's essential to correctly identify your matrix elements. Let's take the example matrix \[ \begin{bmatrix} -1 & -7 \ -5 & 9 \end{bmatrix} \]. Each element has its place:

• \( a = -1 \) — top-left element.
• \( b = -7 \) — top-right element.
• \( c = -5 \) — bottom-left element.
• \( d = 9 \) — bottom-right element.

It is crucial to keep these elements straight as errors in identification can lead to incorrect calculations. Accurate identification of these elements allows you to confidently move onto performing algebraic operations with them. Recognizing their positions in the matrix ensures the correct application of the determinant formula.

Performing Algebraic Calculation

Algebraic calculation is the process of manipulating numbers and variables to arrive at an answer. When calculating the determinant of a matrix, algebraic skills are essential to carry out the formula steps correctly.In our example, you would substitute the identified values into the determinant formula: \[ \text{det}(A) = (-1)(9) - (-7)(-5) \] This breakdown involves two main calculations:- First, calculate \( (-1) \times 9 = -9 \).- Second, calculate \( (-7) \times (-5) = 35 \).Finally, subtract the second product from the first: \[ -9 - 35 = -44 \]Thus, the determinant of the given matrix is \(-44\). Breaking down each arithmetic step ensures accuracy and provides a clearer picture of the process. Developing these skills in algebra enhances one's ability to perform more complex operations with matrices and other mathematical structures.