Matrix inverse (inv) in MATLAB - Step-by-Step Execution

Concept Flow - Matrix inverse (inv)

Start with matrix A

↓

Check if A is square

Yes↓

Check if A is invertible (det(A) != 0)

Yes↓

Calculate inverse using inv(A)

↓

Output inverse matrix A_inv

↓

End

The process starts with a square matrix A, checks if it is invertible, then calculates and outputs its inverse.

Execution Sample

MATLAB

A = [1 2; 3 4];
A_inv = inv(A);
disp(A_inv);

This code calculates and displays the inverse of a 2x2 matrix A.

Execution Table

Step	Action	Evaluation	Result
1	Define matrix A	A = [1 2; 3 4]	[1 2; 3 4]
2	Check if A is square	size(A) = 2x2	True
3	Calculate determinant det(A)	det(A) = 1*4 - 2*3	-2
4	Check if det(A) != 0	-2 != 0	True (invertible)
5	Calculate inverse A_inv = inv(A)	inv(A) = (1/det(A))*[4 -2; -3 1]	[-2 1; 1.5 -0.5]
6	Display A_inv	disp(A_inv)	[-2 1; 1.5 -0.5]

💡 Matrix is square and invertible, inverse calculated successfully.

Variable Tracker

Variable	Start	After Step 1	After Step 3	After Step 5	Final
A	undefined	[1 2; 3 4]	[1 2; 3 4]	[1 2; 3 4]	[1 2; 3 4]
detA	undefined	undefined	-2	-2	-2
A_inv	undefined	undefined	undefined	[-2 1; 1.5 -0.5]	[-2 1; 1.5 -0.5]

Key Moments - 3 Insights

Why must the matrix be square to find its inverse?

Only square matrices have inverses because the inverse operation requires the matrix to have the same number of rows and columns, as shown in Step 2 of the execution_table.

What happens if the determinant is zero?

If the determinant is zero, the matrix is not invertible (singular), so inv(A) cannot be calculated. This is checked in Step 4.

How is the inverse matrix calculated for a 2x2 matrix?

For a 2x2 matrix, the inverse is (1/det(A)) times the matrix with swapped diagonal elements and negated off-diagonal elements, as shown in Step 5.

Visual Quiz - 3 Questions

Look at the execution_table, what is the determinant of matrix A at Step 3?

A 2

B 0

C -2

D 1

💡 Hint

Check the 'Evaluation' column at Step 3 in the execution_table.

At which step does the program confirm that the matrix is invertible?

A Step 2

B Step 4

C Step 5

D Step 6

💡 Hint

Look for the check of determinant not equal to zero in the execution_table.

If matrix A was not square, what would happen according to the concept_flow?

A The program would stop before calculating inverse

B The inverse would still be calculated

C The determinant would be zero

D The inverse would be an identity matrix

💡 Hint

Refer to the decision after 'Check if A is square' in the concept_flow diagram.

Concept Snapshot

Matrix inverse (inv) in MATLAB:
- Use inv(A) to find inverse of square matrix A
- Matrix must be square and det(A) != 0
- For 2x2: inv(A) = (1/det(A))*[d -b; -c a]
- Non-invertible if determinant is zero
- inv(A)*A = Identity matrix

Full Transcript

This visual trace shows how MATLAB calculates the inverse of a matrix using the inv function. First, the matrix A is defined. Then, the program checks if A is square, which is necessary for inversion. Next, it calculates the determinant of A. If the determinant is not zero, the matrix is invertible. The inverse is then calculated using the formula for inv(A). Finally, the inverse matrix is displayed. Key points include the requirement for a square matrix and a non-zero determinant to compute the inverse successfully.