Inverse Matrix Formula Explained with Examples (Adjoint Method)

The inverse matrix formula in plain words

Two formulas do most of the work when you study inverse matrices. The first is for 2 x 2 matrices and is short enough to learn in a minute. The second is the general rule, A^-1 = adj A / det A, which works for any square matrix that has an inverse.

This article explains each formula, shows what the symbols mean, and applies them to Indian examples such as shop prices and investment splits. If you want to test a formula on your own numbers, the inverse matrix calculator gives a quick check.

Where the formula comes from

The inverse of a matrix A is the matrix A^-1 that gives A x A^-1 = A^-1 x A = I, where I is the identity matrix. It plays the same role as the reciprocal in ordinary numbers. The reciprocal of 5 is 1/5, because 5 x 1/5 = 1. In the same way, A^-1 undoes the effect of A.

The Class 12 NCERT book on Determinants builds the general formula from two ideas, the adjoint and the determinant. You can read the chapter for free on ncert.nic.in.

Formula 1: The 2 x 2 inverse

For A = [[a, b], [c, d]]:

A^-1 = (1 / (ad - bc)) x [[d, -b], [-c, a]]

The number ad – bc is the determinant. It must not be zero. The matrix [[d, -b], [-c, a]] is the adjoint. You get it by swapping the diagonal entries and changing the signs of the other two.

Formula 2: The general rule

For any square matrix A with det A not equal to zero:

A^-1 = (1 / det A) x adj A

Here adj A is the transpose of the cofactor matrix. The cofactor of the entry in row i and column j is (-1)^(i+j) x M(ij), where M(ij) is the minor, the determinant of the smaller matrix left after removing row i and column j.

Key takeaway: The whole inverse formula is just three things in order: the determinant, the adjoint and a division. Learn them as a chain and you will not forget any part.

Useful properties and related formulas

Property	Formula	In plain words
Inverse of inverse	(A^-1)^-1 = A	Undoing the undo gives the start
Product rule	(AB)^-1 = B^-1 A^-1	Order reverses
Transpose rule	(A^T)^-1 = (A^-1)^T	You can swap the two operations
Determinant of inverse	det(A^-1) = 1 / det A	Determinants flip to reciprocals
Adjoint relation	A x adj A = (det A) x I	Gives the inverse formula
Solving equations	If AX = B, then X = A^-1 B	Multiply both sides by the inverse

The product rule trips up many students. Think of putting on socks and then shoes. To undo it, you remove the shoes first and the socks second. So the order reverses.

Worked example 1: Applying the 2 x 2 formula

Find the inverse of A = [[5, 3], [2, 1]].

Step 1: Here a = 5, b = 3, c = 2 and d = 1.
Step 2: det A = ad – bc = (5)(1) – (3)(2) = 5 – 6 = -1.
Step 3: The adjoint is [[d, -b], [-c, a]] = [[1, -3], [-2, 5]].
Step 4: A^-1 = (1 / -1) x [[1, -3], [-2, 5]] = [[-1, 3], [2, -5]].

Check: A x A^-1: first row first column is 5(-1) + 3(2) = -5 + 6 = 1. First row second column is 5(3) + 3(-5) = 15 – 15 = 0. Second row first column is 2(-1) + 1(2) = 0. Second row second column is 2(3) + 1(-5) = 1. We get the identity matrix. Note that a negative determinant is fine. It just flips all the signs.

Worked example 2: Prices at a kirana store

A shopkeeper in Nagpur sells rice and dal. Customer one buys 3 kg of rice and 2 kg of dal and pays Rs 390. Customer two buys 1 kg of rice and 4 kg of dal and pays Rs 530. Find the price per kg of each. Let rice cost x and dal cost y.

The equations are 3x + 2y = 390 and x + 4y = 530. In matrix form, AX = B with A = [[3, 2], [1, 4]], X = [[x], [y]] and B = [[390], [530]].

Step 1: Determinant. det A = (3)(4) – (2)(1) = 12 – 2 = 10.
Step 2: Inverse. A^-1 = (1/10) x [[4, -2], [-1, 3]].
Step 3: Multiply. X = A^-1 B = (1/10) x [[4(390) – 2(530)], [-1(390) + 3(530)]].
First entry: 1560 – 1060 = 500. Second entry: -390 + 1590 = 1200.
Step 4: Divide by 10. x = 50 and y = 120.

Answer: Rice is Rs 50 per kg and dal is Rs 120 per kg. Check: 3(50) + 2(120) = 150 + 240 = 390, and 50 + 4(120) = 530. Both are correct. If you prefer to see other ways of solving the same pair, an simultaneous equations solver gives the same result.

Worked example 3: Splitting an investment

Sunita has Rs 5,00,000 to invest. She puts some in a scheme that pays 6 percent a year and the rest in another that pays 9 percent. Her total yearly income is Rs 36,000. How much is in each scheme? (The rates here are only for practice. Real rates should be taken from official bank or post office sources.)

Let x be the amount at 6 percent and y the amount at 9 percent.
Equation 1: x + y = 500000.
Equation 2: 0.06x + 0.09y = 36000.

So A = [[1, 1], [0.06, 0.09]] and B = [[500000], [36000]].

Determinant: (1)(0.09) – (1)(0.06) = 0.03.
Inverse: A^-1 = (1 / 0.03) x [[0.09, -1], [-0.06, 1]].
x: (0.09 x 500000 – 1 x 36000) / 0.03 = (45000 – 36000) / 0.03 = 9000 / 0.03 = 300000.
y: (-0.06 x 500000 + 36000) / 0.03 = (-30000 + 36000) / 0.03 = 6000 / 0.03 = 200000.

Answer: Rs 3,00,000 is at 6 percent and Rs 2,00,000 is at 9 percent. Check: 18,000 + 18,000 = Rs 36,000. This is a good example of how the inverse formula answers a money question.

Benefits of mastering the formula

A clear formula removes guesswork. You do not need to invent a new method for each question, because the same three steps apply every time. The formula also links several topics. Determinants, cofactors, adjoints and linear equations all meet in one place, so revising one helps with all. In exams, a correct formula with neat steps earns method marks even when the final number is off. In real work, the idea behind the formula supports computer graphics, data fitting and economic models, so the effort pays off beyond school.

Challenges and limitations

The formula is exact, but it is not always practical. For a 4 x 4 matrix, you need sixteen cofactors, each a 3 x 3 determinant. That is too slow by hand, and software uses faster methods instead.

The formula also needs a non-zero determinant. When the determinant is zero, the matrix is singular and there is no inverse at all. In practice, a determinant very close to zero may give unstable answers when numbers are rounded, which is something engineers watch for.

Common mistakes to avoid

1. Using ad + bc instead of ad – bc. The determinant is a difference, not a sum. Write it as “main diagonal minus other diagonal”.
2. Changing the wrong signs. In the 2 x 2 adjoint, swap a and d, and flip only b and c.
3. Using (AB)^-1 = A^-1 B^-1. The correct rule reverses the order: B^-1 A^-1.
4. Multiplying in the wrong order when solving AX = B. The inverse goes on the left: X = A^-1 B, not B A^-1.
5. Forgetting the minus signs in cofactors. Use the + – + checkerboard pattern and write the sign beside each minor.
6. Not testing the final matrix. Multiply A by A^-1. If you do not get the identity matrix, find the slip before moving on.

Best practices and expert recommendations

1. Memorise the 2 x 2 rule as a picture. Imagine swapping the main diagonal and flipping the signs on the other one. A picture lasts longer than words.
2. Write the chain on paper first. Put det, adj and divide at the top of the page before you start a 3 x 3 problem.
3. Verify with a quick product. Check one row and one column. If both are right, the rest is likely right too.
4. Keep exact fractions. Use 1/10 instead of 0.1 where possible, so you avoid rounding drift in longer working.
5. Link the formula to a story. Prices, mixtures and investments make the equations meaningful and easier to recall in the exam hall.
6. Use official practice sets. NCERT exercises on ncert.nic.in and CBSE sample papers on cbse.gov.in match the exam level, and a matrix solver can confirm your answers after you finish.

Try it yourself

Use the formula to find the inverse of [[2, 1], [7, 4]]. The determinant is 8 – 7 = 1, so the inverse is [[4, -1], [-7, 2]]. Then type the same matrix into our free online inverse matrix calculator and compare. Next, change one entry to make the determinant zero and see what the tool tells you.

Conclusion

The inverse matrix formula is A^-1 = adj A / det A, with a handy shortcut for 2 x 2 matrices. Learn the determinant, the adjoint and the final division as one chain. Remember the product rule with reversed order, check every answer by multiplying back, and practise with money-based problems from daily Indian life. That is enough to handle most school and entrance questions on this topic.

FAQs

What is the formula for the inverse of a matrix?
The formula is A^-1 = adj A divided by det A. The determinant must not be zero. For a 2×2 matrix, you can swap the diagonal, flip the signs of the other two entries and divide by ad – bc.

Why must the determinant be non-zero?
The formula divides by the determinant. Division by zero is not possible, so a matrix with a zero determinant has no inverse.

What is the rule for the inverse of a product of matrices?
The inverse of a product reverses the order: (AB)^-1 = B^-1 A^-1. Both A and B must be invertible.

How do I solve linear equations using the inverse?
Write the system as AX = B. If A has an inverse, then X = A^-1 B. Multiply the inverse by B in that order.

Can the inverse formula be used for any size of matrix?
Yes, the adjoint formula works for any square size with a non-zero determinant. For matrices bigger than 3×3, row operations are much faster than cofactors.