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Chapter 6: Problem 7

Evaluate each determinant. $$\left|\begin{array}{ll}-5 & -1 \\\\-2 & -7\end{array}\right|$$

Short Answer

Expert verified
The determinant of the matrix is 33.

Step by step solution

01

Identify the elements of the matrix

The matrix here is \( \begin{bmatrix}-5 & -1 \\-2 & -7\end{bmatrix}\), where \( a = -5, b = -1, c = -2, d = -7\).
02

Substitute the values into the formula

Follow the formula \(ad - bc\) and substitute the values from the matrix into the formula, which gives \((-5)*(-7) - (-1)*(-2)\).
03

Do the multiplication and subtraction

Begin by multiplying the numbers to get \(35 - 2\). Then, subtract 2 from 35 to get 33.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Formation
Matrices are essentially rectangular arrays of numbers arranged in rows and columns. They are a key tool in linear algebra, providing a compact way to represent systems of equations and perform various operations. In this exercise, the given matrix is a 2x2 matrix:
  • The first row contains the numbers -5 and -1.
  • The second row contains the numbers -2 and -7.
This particular structure allows us to easily access and identify individual elements, which is crucial for further calculations like finding the determinant.
Determinant Formula
The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix, such as whether it is invertible or not. For a 2x2 matrix, the determinant is calculated using the formula:\[ ext{det}(A) = ad - bc\]where:
  • \(a, b, c,\) and \(d\) are elements of the matrix
In this problem, the determinant of the matrix \( \begin{bmatrix}-5 & -1 \ -2 & -7 \end{bmatrix} \) is found by substituting the matrix elements into the formula. This involves calculating \((-5)(-7) - (-1)(-2)\).
Matrix Multiplication
Matrix multiplication involves multiplying elements from the rows of the first matrix by elements from the columns of the second matrix. However, when determining a 2x2 matrix's determinant, multiplication refers specifically to the terms in the determinant formula. In the step-by-step solution:
  • The multiplication of \( a \) and \( d \) was performed: \((-5) \times (-7) = 35\).
  • Similarly, the product of \( b \) and \( c \) was calculated: \((-1) \times (-2) = 2\).
Matrix multiplication in this context is about understanding how numbers inside the matrix interact, which provides you the terms needed for further operations.
Matrix Subtraction
After matrix multiplication within the determinant formula, subtraction unites these results to give the final determinant value. In a 2x2 matrix, subtraction occurs between the products evaluated in the previous step.Here's what happened:
  • After multiplying, we had 35 from the product of \( a \) and \( d \), and 2 from the product of \( b \) and \( c \).
  • To complete the determinant evaluation, we subtracted the second product from the first, giving us \( 35 - 2 = 33\).
By following this step carefully, you can always find the determinant of any 2x2 matrix effectively.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Find two matrices \(A\) and \(B\) such that \(A B-B A\).

The figure shows the letter \(L\) in a rectangular coordinate system. (GRAPH CANNOT COPY) The figure can be represented by the matrix $$B=\left[\begin{array}{llllll}0 & 3 & 3 & 1 & 1 & 0 \\\0 & 0 & 1 & 1 & 5 & 5\end{array}\right]$$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The \(L\) is completed by connecting the last point in the matrix, \((0,5),\) to the starting point, \((0,0) .\) Use these ideas to solve Exercises \(53-60 .\) Use matrix operations to move the L 2 units to the right and 3 units down. Then graph the letter and its transformation in a rectangular coordinate system.

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows If an operation is not defined, state the reason. $$A=\left[\begin{array}{rr}4 & 0 \\\\-3 & 5 \\\0 & 1\end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\\\-2 & -2\end{array}\right] \quad C=\left[\begin{array}{rr}1 & -1 \\\\-1 & 1\end{array}\right]$$ $$A-C$$

What is a cryptogram?

Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) is $$\text { Area }-\pm \frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1 \\\x_{3} & y_{3} & 1\end{array}\right|$$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use determinants to find the area of the triangle whose vertices are \((1,1),(-2,-3),\) and \((11,-3)\)
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