Math Problem Statement

Use determinants to find out if the matrix is invertible.

Solution

To determine if a matrix is invertible, we need to calculate its determinant. If the determinant of a matrix is non-zero, then the matrix is invertible. For the given 3×33 \times 3 matrix:

[104321260209]\begin{bmatrix} 10 & 4 & 3 \\ 2 & -12 & 6 \\ 0 & 20 & -9 \end{bmatrix}

we'll calculate the determinant by expanding along the first row.

Step-by-Step Determinant Calculation

The determinant of a 3×33 \times 3 matrix [abcdefghi]\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} is calculated as:

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

For our matrix:

  • a=10a = 10, b=4b = 4, c=3c = 3
  • d=2d = 2, e=12e = -12, f=6f = 6
  • g=0g = 0, h=20h = 20, i=9i = -9

Substitute these values into the determinant formula:

det=10((12)(9)(6)(20))4((2)(9)(6)(0))+3((2)(20)(12)(0))\text{det} = 10((-12)(-9) - (6)(20)) - 4((2)(-9) - (6)(0)) + 3((2)(20) - (-12)(0))

Calculate Each Term

  1. For the first term: 10×((12)(9)(6)(20))10 \times ((-12)(-9) - (6)(20))

    • (12)(9)=108(-12)(-9) = 108
    • (6)(20)=120(6)(20) = 120
    • So, 10×(108120)=10×(12)=12010 \times (108 - 120) = 10 \times (-12) = -120
  2. For the second term: 4×((2)(9)(6)(0))-4 \times ((2)(-9) - (6)(0))

    • (2)(9)=18(2)(-9) = -18
    • (6)(0)=0(6)(0) = 0
    • So, 4×(18)=72-4 \times (-18) = 72
  3. For the third term: 3×((2)(20)(12)(0))3 \times ((2)(20) - (-12)(0))

    • (2)(20)=40(2)(20) = 40
    • (12)(0)=0(-12)(0) = 0
    • So, 3×40=1203 \times 40 = 120

Combine the Terms

det=120+72+120=72\text{det} = -120 + 72 + 120 = 72

Since the determinant is 7272, which is not zero, the matrix is invertible.

Would you like further details on any steps, or do you have any questions?


Related Questions

  1. What does it mean if a matrix has a determinant of zero?
  2. How do you calculate the inverse of a 3×33 \times 3 matrix?
  3. Can a non-square matrix be invertible?
  4. How does the determinant change if we swap two rows in a matrix?
  5. What is the geometric interpretation of a matrix's determinant?

Tip

For larger matrices, using row or column expansion with the most zeros simplifies determinant calculations significantly.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Determinants
Matrix Inversion

Formulas

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

Theorems

Invertibility of a Matrix (A matrix is invertible if its determinant is non-zero)

Suitable Grade Level

Grades 11-12