The inverse of 3 x 3 matrix with determinants and adjugate

The inverse of 3 x 3 matrix with determinants and adjugate

In this lesson, you will learn the long way of computing the inverse of a 3 x 3 matrix. This method requires 4 steps. The first step is the matrix of minor. Each entry in the matrix is a 2 x 2 matrix that is not in that entry's row or column. The second step is the adjugate. This requires you to take your matrix of minors and changing the signs of certain entries depending on the negative signs that appear in the checkerboard. The third step is to transpose. This requires you to switch all the rows and make them into columns. The last step is to multiply your transposed matrix by 1 over the determinant of the original matrix (scalar multiplication). All of these steps should now give you the inverse.
Basic concepts: The determinant of a 2 x 2 matrix, The determinant of a 3 x 3 matrix (General & Shortcut Method),
Related concepts: The three types of matrix row operations,

Lessons

This method is the long way of computing the inverse of a 3 x 3 matrix. To do this, we need to go through 4 steps:

1) The Matrix of Minors
2) The Adjugate
3) Transpose
4) Multiply by 1determinantoforiginalmatrix\;\frac{1}{determinant\; of\; original\; matrix}

Once we apply these steps, then we will find the inverse.
  • Introduction
    The Inverse of 3 x 3 Matrix Overview:
    a)
    The Matrix of Minors

    b)
    The Adjugate Matrix

    c)
    Transpose

    d)
    Multiply by 1determinantoforiginalmatrix\;\frac{1}{determinant\; of\; original\; matrix}

  • 1.
    Finding the Matrix of Minors
    You are given that The inverse of 3 x 3 matrix with determinants and adjugate. Find the Matrix of Minors
  • 2.
    Finding the Adjugate Matrix
    You are given that the matrix of minors is The inverse of 3 x 3 matrix with determinants and adjugate. Find the Adjugate matrix.
  • 3.
    Transposing the Adjugate
    You are given that the Adjugate Matrix is The inverse of 3 x 3 matrix with determinants and adjugate. Transpose this matrix.
  • 4.
    Getting the inverse
    You are given that The inverse of 3 x 3 matrix with determinants and adjugate. The transposed adjugate of this matrix is The inverse of 3 x 3 matrix with determinants and adjugate. Find the inverse of AA.
  • 5.
    Applying the 4 steps to get the inverse
    You are given that The inverse of 3 x 3 matrix with determinants and adjugate. Find the inverse of this matrix
  • 6.
    Applying the 4 steps to get the inverse
    You are given that The inverse of 3 x 3 matrix with determinants and adjugate. Find the inverse of this matrix
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48.
Determinants and Inverses of Matrices
48.1
The determinant of a 2 x 2 matrix
48.2
The determinant of a 3 x 3 matrix (General & Shortcut Method)
48.3
The inverse of a 2 x 2 matrix
48.4
The inverse of 3 x 3 matrices with matrix row operations
48.5
The inverse of 3 x 3 matrix with determinants and adjugate
48.6
2 x 2 invertible matrix
48.7
Solving linear systems using Cramer's Rule
48.8
Solving linear systems using 2 x 2 inverse matrices
ACCUPLACER Test Prep