[[-2,4,-5],[1,3,-7],[-1,4,-8]] find the det

To find the determinant of the given \( 3 \times 3 \) matrix:

\[
\text{Matrix} = \begin{bmatrix} -2 & 4 & -5 \\ 1 & 3 & -7 \\ -1 & 4 & -8 \end{bmatrix}
\]

We use the formula for the determinant of a \( 3 \times 3 \) matrix:

\[
\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
\]

Where the matrix elements are:

\[
\text{Matrix} = \begin{bmatrix} 
a & b & c \\ 
d & e & f \\ 
g & h & i 
\end{bmatrix} = \begin{bmatrix} 
-2 & 4 & -5 \\ 
1 & 3 & -7 \\ 
-1 & 4 & -8 
\end{bmatrix}
\]

Substituting the values:

\[
\text{det}(A) = (-2)((3)(-8) - (-7)(4)) - 4((1)(-8) - (-7)(-1)) + (-5)((1)(4) - (3)(-1))
\]

Now, let's compute the terms:

1. \( (3)(-8) - (-7)(4) = -24 + 28 = 4 \)
2. \( (1)(-8) - (-7)(-1) = -8 - 7 = -15 \)
3. \( (1)(4) - (3)(-1) = 4 + 3 = 7 \)

So, the determinant is:

\[
\text{det}(A) = -2(4) - 4(-15) - 5(7)
\]

\[
= -8 + 60 - 35 = 17
\]

Therefore, the determinant of the matrix is:

\[
\boxed{17}
\]

Would you like more details on any specific part of this calculation?

---

Here are five related questions you might find helpful:

1. How do you calculate the determinant of a \(2 \times 2\) matrix?
2. What is the significance of the determinant of a matrix in linear algebra?
3. Can you find the determinant of a larger matrix, such as \(4 \times 4\)?
4. How can you use determinants to solve systems of linear equations?
5. What does it mean if the determinant of a matrix is zero?

**Tip:** When calculating determinants of matrices, breaking down the computation step by step can help avoid errors.

Learn how to calculate the determinant of a 3x3 matrix with a detailed step-by-step guide. This problem involves matrix algebra and determinants, covering the formula for the determinant of a 3x3 matrix. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation