Inverse of a Matrix and Linear Systems: Quick Revision

A brief note on Inverse of a Matrix and Linear Systems. Read it before a test. Try each question at the end before you open its answer.

The adjoint and the inverse

• Inverse: $AA^{-1} = A^{-1}A = I$. It exists exactly when $|A| \ne 0$. A matrix with $|A| = 0$ is singular.
• Adjoint: the transpose of the cofactor matrix, and $A\,\text{adj}\,A = |A|I$.
• Formula: $A^{-1} = \dfrac{\text{adj}\,A}{|A|}$.
• Order 2: swap the diagonal entries and negate the other two, then divide by $ad - bc$.
• Rules: $(AB)^{-1} = B^{-1}A^{-1}$, $(A^T)^{-1} = (A^{-1})^T$, $|A^{-1}| = \dfrac{1}{|A|}$ and $|\text{adj}\,A| = |A|^{n-1}$.

Linear systems

• Matrix form: $AX = B$. If $|A| \ne 0$, the unique solution is $X = A^{-1}B$.
• If $|A| = 0$: there is no solution or infinitely many. Compare the equations.
• Homogeneous: $AX = O$ has a non-zero solution exactly when $|A| = 0$.
• Cramer's rule: $x_i = \Delta_i/\Delta$, where $\Delta_i$ has column $i$ replaced by $B$.
• Rows: a row $(0\ 0\ 0 \,|\, c)$ with $c \ne 0$ means no solution. A row $(0\ 0\ 0 \,|\, 0)$ means a free unknown.

Row operations

• The three operations: swap two rows, scale a row by a non-zero number, add a multiple of one row to another.
• The inverse: $[A \,|\, I] \to [I \,|\, A^{-1}]$.
• Models: flow in equals flow out at a junction, and an economy gives $x = (I - C)^{-1}d$.

A worked example

Solve $3x + y = 7$ and $5x + 2y = 12$ with the inverse.

1. The matrix. $A =

$,$B =

$, and$|A| = 6 - 5 = 1$.

2. The inverse. $A^{-1} =

$.

3. The solution. $X = A^{-1}B =

=

$. Check:$3 \cdot 2 + 1 = 7$.

Practice with new numbers

A program to try

The program checks the rules of this note on one matrix: $A\,\text{adj}\,A = |A|I$, the inverse, the inverse of a product and a solution of $AX = B$.

Mistakes to avoid

• Forgetting to transpose the cofactors. The adjoint is the transpose.
• Not dividing by $|A|$. Only when $|A| = 1$ is the adjoint the inverse.
• Writing $(AB)^{-1} = A^{-1}B^{-1}$. The order reverses.
• Calling $|A| = 0$ always "no solution". It may be infinitely many.
• Writing $X = BA^{-1}$ for $AX = B$. It is $X = A^{-1}B$.

Check yourself

1. Find the inverse of $

$.

$.

2. $A$ is $3 \times 3$ with $|A| = 2$. Find $|\text{adj}\,A|$.

3. Does $x + y = 4$, $2x + 2y = 9$ have a solution?

4. Solve $x + y = 5$, $x - y = 1$ with Cramer's rule.

5. After row operations the last row of $[A \,|\, B]$ is $(0\ 0\ 0 \,|\, 0)$. What follows?

Sources

• NCERT, Mathematics Textbook for Class XII, Part I, Chapters 3 and 4 (Matrices, Determinants).
• OpenStax, Precalculus 2e, Sections 9.6 to 9.8 (Gaussian elimination, inverses, Cramer's rule).