Math Problem Statement

Given matrix A = [ [1, -2, 0], [3, 1, 5], [-1, 2, 3] ], find det(A).

Solution

To find the determinant of a 3x3 matrix, AA, we can use the formula:

det(A)=a(eifh)b(difg)+c(dheg)\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

where the matrix AA is:

A=(abcdefghi)A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}

For the matrix given in your image:

A=(120315123)A = \begin{pmatrix} 1 & -2 & 0 \\ 3 & 1 & 5 \\ -1 & 2 & 3 \end{pmatrix}

Let's identify each element in terms of a,b,c,d,a, b, c, d, etc.:

  • a=1a = 1
  • b=2b = -2
  • c=0c = 0
  • d=3d = 3
  • e=1e = 1
  • f=5f = 5
  • g=1g = -1
  • h=2h = 2
  • i=3i = 3

Step-by-Step Determinant Calculation:

det(A)=1(1352)(2)(335(1))+0(321(1))\text{det}(A) = 1 \cdot (1 \cdot 3 - 5 \cdot 2) - (-2) \cdot (3 \cdot 3 - 5 \cdot (-1)) + 0 \cdot (3 \cdot 2 - 1 \cdot (-1))

Breaking it down:

  1. First term: 1(1352)=1(310)=1(7)=71 \cdot (1 \cdot 3 - 5 \cdot 2) = 1 \cdot (3 - 10) = 1 \cdot (-7) = -7
  2. Second term: (2)(335(1))=2(9+5)=214=28-(-2) \cdot (3 \cdot 3 - 5 \cdot (-1)) = 2 \cdot (9 + 5) = 2 \cdot 14 = 28
  3. Third term: Since the third term has c=0c = 0, it contributes 00 to the determinant.

Adding these results:

det(A)=7+28+0=21\text{det}(A) = -7 + 28 + 0 = 21

Answer:

det(A)=21\text{det}(A) = 21

Would you like further details or have any questions?


Here are 5 related questions that expand on this topic:

  1. How would the determinant calculation change if one of the entries in matrix AA were different?
  2. What does the determinant of a matrix tell us about the matrix?
  3. How can we determine if a 3x3 matrix is invertible using its determinant?
  4. How would we calculate the determinant of a 4x4 matrix?
  5. Can the determinant be used to find the area or volume spanned by vectors?

Tip: The sign of each term in the cofactor expansion alternates. For a 3x3 matrix, it follows the pattern +,,++, -, + for each element in the first row.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Determinants

Formulas

Determinant of 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Theorems

Cofactor Expansion for Determinants

Suitable Grade Level

Grades 10-12