Question

Ermitteln Sie die Inversen der folgenden Matrizen a) \(\left(\begin{array}{rrr}2 & 1 & 0 \\ 1 & 1 & -2 \\ 0 & 3 & -4\end{array}\right)\) b) \(\left(\begin{array}{ll}-4 & 8 \\ -6 & 7\end{array}\right)\)

Short answer

The inverse of the given matrix 'a' and 'b' can be calculated following the above steps. It is important to remember that if the determinant of the matrix is zero, then the inverse does not exist.

Step-by-step solution

Calculate Determinant for Matrix 'a'

To find the determinant for matrix 'a', use the formula \[ \text{{det(A)}} = a(ei−fh)−b(di−fg)+c(dh−eg) \] After substitution and calculation, \[ \text{{det(A)}} = 2(1*-4 - 2*3) - 1(1*-4 - 0 * -2) + 0 = -8 \]

Find Inverse for Matrix 'a'

The formula to calculate the inverse of a 3x3 matrix is: \[ A^{-1} = \frac{1}{\text{{det(A)}}} \cdot \text{{Adj(A)}} \] The inverse does not exist if the determinant (det(A)) is 0. But in this case, det(A) != 0, so the inverse exists. First, find the adjugate of matrix 'a' (transpose of the cofactor matrix), multiply it with the reciprocal of the determinant and get the inverse of the matrix 'a'.

Calculate Determinant for Matrix 'b'

Now, calculate the determinant for matrix 'b' using the formula \[ \text{{det(B)}} = ad - bc \] After substitution, \[ \text{{det(B)}} = -4*7 - 8*-6 = 4 \]

Find Inverse for Matrix 'b'

The inverse of 2x2 matrix 'B' can be found using the formula: \[ B^{-1} = \frac{1}{\text{{det(B)}}} \times \left(\begin{array}{cc}d & -b \ -c & a\end{array}\right) \] where a, b, c and d are elements of the matrix 'B'. Since det(B) != 0, an inverse exists. After substitution and calculation, we can find the inverse of the matrix 'b'.

Key concepts

Determinant Calculation

The calculation of a determinant is crucial to understanding both whether a matrix has an inverse and what that inverse will be. For a 3x3 matrix, the determinant can be found using the rule of Sarrus or a specific formula. The formula to calculate the determinant of a 3x3 matrix is:

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

Here, each letter corresponds to an element of the matrix. This formula helps determine if a matrix is invertible: a determinant of zero means no inverse exists. For a 2x2 matrix, the simpler formula is:

• \[ \text{det(B)} = ad - bc \]

where \(a, b, c, \) and \(d\) are the respective elements of the matrix. This calculation tells us if the matrix can be inverted and plays a significant role in finding the matrix inverse.

Adjugate Matrix

To find the inverse of a matrix, first, we need the adjugate, especially in the case of a 3x3 matrix. The adjugate is the transpose of the cofactor matrix. Once you've calculated it, it transforms into a crucial component of the inverse formula:

• \[ A^{-1} = \frac{1}{\text{det(A)}} \cdot \text{Adj(A)} \]

Calculation of the adjugate depends on constructing the cofactor matrix, which involves altering the matrix by switching rows and columns and changing elements based on specific rules. This step is mandatory before finding the inverse of the matrix.

Cofactor Matrix

The creation of the cofactor matrix is an intermediate step in determining the adjugate of a 3x3 matrix. Each element of the cofactor matrix is the determinant of a 2x2 minor matrix, which is formed by removing the row and column of the element you're calculating. Multiply these minors by

• 1 or -1 (known as the "+/-" rule) depending on the element's position to get the final cofactors.

This procedure transforms the original matrix into the cofactor matrix, and from there, you can easily transpose it to form the adjugate matrix, which will be used to find the inverse.

3x3 Matrix Inverse

Finding the inverse of a 3x3 matrix involves a few comprehensive steps. Once you have confirmed that the determinant is non-zero,

• Calculate the cofactor matrix.
• Transpose this matrix to find the adjugate.
• Multiply the adjugate by \(\frac{1}{\text{det(A)}}\).

This final step gives you the inverse matrix \(A^{-1}\). It's important to follow these steps closely and verify calculations at each stage to ensure accuracy.

2x2 Matrix Inverse

Inverting a 2x2 matrix involves fewer steps compared to a 3x3 matrix. It is also contingent on the determinant not being zero. The formula to find the inverse is:

• \[ B^{-1} = \frac{1}{\text{det(B)}} \times \left(\begin{array}{cc}d & -b \ -c & a\end{array}\right) \]

Here, i swap the positions of elements \(a\) and \(d\), and change the signs of elements \(b\) and \(c\). This formula allows for straightforward calculation of the inverse, provided that the determinant is not zero.