3x3 Matrix Inverse Calculator

Enter Matrix A:
Matrix Visualization

Calculation Steps

Formulas:

1. Determinant of a 3x3 matrix:

\[|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]

2. Adjoint matrix:

\[adj(A) = \begin{bmatrix} C_{11} & C_{21} & C_{31} \ C_{12} & C_{22} & C_{32} \ C_{13} & C_{23} & C_{33} \end{bmatrix}^T\]

Where \(C_{ij}\) is the cofactor of element \(a_{ij}\).

3. Inverse matrix:

\[A^{-1} = \frac{1}{|A|} adj(A)\]

Step-by-step calculation:

  1. Calculate the determinant: \[|A| = ( \cdot - \cdot ) - ( \cdot - \cdot ) - ( \cdot - \cdot ) = 0.00\]
  2. Calculate the cofactor matrix: \[C_{11} = (-1)^{1+1} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{12} = (-1)^{1+2} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{13} = (-1)^{1+3} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{21} = (-1)^{2+1} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{22} = (-1)^{2+2} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{23} = (-1)^{2+3} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{31} = (-1)^{3+1} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{32} = (-1)^{3+2} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{33} = (-1)^{3+3} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]
  3. Transpose the cofactor matrix to get the adjoint: \[adj(A) = \begin{bmatrix}\end{bmatrix}\]

Verification:

We can verify our result by multiplying A with its inverse (if it exists):

Since the determinant is zero, the matrix is not invertible, and we cannot perform this verification.

3x3 Matrix Inverse, Determinant, and Adjoint

What is a Matrix Inverse?

The inverse of a square matrix A, denoted as A^(-1), is a matrix that, when multiplied with A, results in the identity matrix. For a 3x3 matrix A, if A^(-1) exists:

\[A \cdot A^{-1} = A^{-1} \cdot A = I_3\]

Where I_3 is the 3x3 identity matrix.

Matrix Inverse Formula

For a 3x3 matrix A, its inverse is given by:

\[A^{-1} = \frac{1}{|A|} \cdot adj(A)\]

Where:|A| is the determinant of A adj(A) is the adjoint (adjugate) of A

Determinant of a 3x3 Matrix

The determinant of a 3x3 matrix A = [a_ij] is calculated as:

\[|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]

Adjoint of a 3x3 Matrix

The adjoint of a 3x3 matrix A is the transpose of its cofactor matrix:

\[adj(A) = \begin{bmatrix} C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32} \\ C_{13} & C_{23} & C_{33} \end{bmatrix}\]

Where C_ij is the cofactor of element a_ij.

Calculation Steps

  1. Calculate the determinant of the matrix.
  2. If the determinant is non-zero, proceed; otherwise, the matrix is not invertible.
  3. Calculate the cofactor matrix.
  4. Transpose the cofactor matrix to get the adjoint.
  5. Divide the adjoint by the determinant to get the inverse.

Example

Let's find the inverse of matrix A:

\[A = \begin{bmatrix} 4 & 7 & 2 \\ 2 & 6 & 3 \\ 1 & 5 & 8 \end{bmatrix}\]

1. Calculate the determinant:

\[|A| = 4(6 \cdot 8 - 3 \cdot 5) - 7(2 \cdot 8 - 3 \cdot 1) + 2(2 \cdot 5 - 6 \cdot 1) = 4(48 - 15) - 7(16 - 3) + 2(10 - 6) = 132 - 91 + 8 = 49\]

2. Calculate the cofactor matrix:

\[C = \begin{bmatrix} 33 & -3 & -9 \\ -31 & 30 & -5 \\ 13 & -18 & 10 \end{bmatrix}\]

3. Transpose to get the adjoint:

\[adj(A) = \begin{bmatrix} 33 & -31 & 13 \\ -3 & 30 & -18 \\ -9 & -5 & 10 \end{bmatrix}\]

4. Divide by the determinant to get the inverse:

\[A^{-1} = \frac{1}{49} \begin{bmatrix} 33 & -31 & 13 \\ -3 & 30 & -18 \\ -9 & -5 & 10 \end{bmatrix} = \begin{bmatrix} \frac{33}{49} & -\frac{31}{49} & \frac{13}{49} \\ -\frac{3}{49} & \frac{30}{49} & -\frac{18}{49} \\ -\frac{9}{49} & -\frac{5}{49} & \frac{10}{49} \end{bmatrix}\]

Visual Representation

A adj(A) A^(-1) 4 7 2 2 6 3 1 5 8 33 -31 13 -3 30 -18 -9 -5 10 0.67 -0.63 0.27 -0.06

The Matrix Multiplication Formula

For two 3x3 matrices A and B, their product C = AB is defined as:

\[C_{ij} = \sum_{k=1}^{3} A_{ik} \times B_{kj}\]

Where:

  • \(C_{ij}\) is the element in the i-th row and j-th column of the resulting matrix C
  • \(A_{ik}\) is the element in the i-th row and k-th column of matrix A
  • \(B_{kj}\) is the element in the k-th row and j-th column of matrix B

Calculation Steps

To multiply two 3x3 matrices:

  1. For each element in the resulting matrix:
    • Multiply corresponding elements from the row of A and column of B
    • Sum these products
  2. Repeat this process for all 9 elements of the resulting matrix

Example Calculation

Let's multiply matrix A by matrix B:

\[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, B = \begin{bmatrix} 9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1 \end{bmatrix}\]

Calculating the first element of C:

\[C_{11} = (1 \times 9) + (2 \times 6) + (3 \times 3) = 9 + 12 + 9 = 30\]

Continuing this process for all elements, we get:

\[C = AB = \begin{bmatrix} 30 & 24 & 18 \\ 84 & 69 & 54 \\ 138 & 114 & 90 \end{bmatrix}\]

Visual Representation

A B A × B 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 30 24 18 84 69 54 138 114 90

This diagram visually represents the multiplication of matrix A and matrix B, resulting in A × B.

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