Matrix Formulas Explained with Examples

Formulas are the backbone of matrices and determinants in NCERT Class 12. They are also the part students forget first, because there are many of them and they look alike. This article lists the important ones, explains what each means in plain words, and shows them working on Indian examples with rupees and city sales.

Instead of memorising blindly, try to understand why each formula exists. When you see the idea behind it, the formula is easy to recall in the exam hall.

The Core Matrix Formulas at a Glance

Before the details, here is a quick table. Keep it near your desk while you practise. Each row is explained in the sections that follow, with a worked example where it matters most.

Concept	Formula in words	Use
Determinant 2 x 2	ad – bc	Test for inverse, area
Determinant 3 x 3	Row expansion with cofactors	Inverse, equations
Cofactor	Minor times (-1) raised to (i plus j)	Adjoint
Inverse	adj(A) divided by |A|	Solving AX = B
Product entry	Row i dot column j	Matrix multiplication
Cramer’s rule	x = Dx divided by D	Linear systems

Notice that most formulas depend on the determinant. That is why the determinant is the most tested idea in this chapter. If you master it, the inverse and Cramer’s rule follow naturally.

Determinant Formulas Explained

The determinant is a single number linked to a square matrix. For a 2 x 2 matrix with first row a, b and second row c, d, the formula is |A| = ad – bc. It tells you whether the matrix can be reversed. A zero determinant means the rows are dependent and the matrix is singular.

For a 3 x 3 matrix, expand along a row. Using the first row with entries a1, a2, a3, the determinant is a1 times C1 plus a2 times C2 plus a3 times C3, where C is the cofactor of each entry. The cofactor of the entry in row i and column j is the minor multiplied by (-1) raised to the power i plus j. That power creates the sign pattern plus, minus, plus across the top row.

Useful Determinant Properties

• Swapping two rows changes the sign of the determinant.
• If two rows are identical or proportional, the determinant is zero.
• Multiplying one row by k multiplies the determinant by k.
• The determinant of a product equals the product of the determinants: |AB| = |A| times |B|.
• For an n x n matrix, |kA| equals k raised to n times |A|.

These properties are favourites in JEE Main. Often the question can be solved in two lines if you spot that a row is a multiple of another, with no expansion needed.

Worked Example 1: A 3 x 3 Determinant in a Sweet Shop

A sweet shop in Indore uses three raw materials for three kinds of sweets. The matrix of kilograms used per batch has rows (2, 1, 3), (0, 4, 1) and (1, 2, 5). We want to know whether the three batches can be planned independently, which depends on whether the determinant is zero.

1. Expand along the first row. The entry 2 has minor (4 times 5) minus (1 times 2), which is 18. Its term is 2 times 18, which is 36.
2. The entry 1 has minor (0 times 5) minus (1 times 1), which is -1. The sign is minus, so the term is minus 1 times -1, which is plus 1.
3. The entry 3 has minor (0 times 2) minus (4 times 1), which is -4. The term is 3 times -4, which is -12.
4. Add the terms: 36 plus 1 minus 12 equals 25.

The determinant is 25, which is not zero. The matrix is non-singular, so the resource plan is solvable and has a unique answer.

The Adjoint and Inverse Formula

The inverse of a square matrix A is written A inverse. It satisfies A times A inverse equals the identity matrix I. The formula is A inverse = adj(A) divided by |A|, and it exists only when |A| is not zero.

The adjoint is the transpose of the cofactor matrix. So the steps are fixed: find every cofactor, arrange them in a matrix, transpose it, then divide by the determinant. For a 2 x 2 matrix with entries a, b, c, d, the adjoint is simply d, -b, -c, a. This is the shortcut of swapping the diagonal and changing the other two signs.

Worked Example 2: Rice and Dal Prices

A kirana store in Lucknow sells 3 kg rice with 1 kg dal for Rs. 270. On another day, 5 kg rice with 2 kg dal costs Rs. 480. Find the price per kg of each.

1. Write A with rows (3, 1) and (5, 2). The determinant is 3 times 2 minus 1 times 5, which is 1.
2. The adjoint has rows (2, -1) and (-5, 3). Dividing by 1 leaves it unchanged, so this is A inverse.
3. Multiply by the cost column (270, 480). Rice: 2 times 270 minus 480 equals 60. Dal: -5 times 270 plus 3 times 480 equals 90.

Rice costs Rs. 60 per kg and dal costs Rs. 90 per kg. Verify: 3 times 60 plus 90 is 270, and 5 times 60 plus 2 times 90 is 480. Both agree.

Expert insight: A matrix whose determinant is 1 has an inverse with whole-number entries if the original has whole numbers. Exam setters often choose such matrices so that you can focus on the method rather than on fractions.

The Matrix Product Formula

If C equals AB, then each entry c_ij is the sum of a_ik times b_kj over all k. In plain words, take row i of A and column j of B, multiply matching entries, and add. The product is defined only when the columns of A equal the rows of B.

Some key product rules are worth knowing. Multiplication is associative, so (AB)C equals A(BC). It is distributive, so A(B plus C) equals AB plus AC. It is not commutative. The identity matrix leaves any matrix unchanged: AI equals IA equals A.

Cramer’s Rule for Linear Equations

Cramer’s rule solves a system of linear equations using determinants. Let D be the determinant of the coefficient matrix. Let Dx be the determinant you get when you replace the x column with the constants. Then x equals Dx divided by D. The same pattern holds for y and z.

The rule works only when D is not zero. If D is zero, the system has either no solution or infinitely many, and you must check further using the adjoint method or row reduction. This is a standard case in NCERT Class 12 and appears often in board papers.

Worked Example 3: Machine Prices Across Three Cities

A company sells three machines A, B and C. A buyer in Mumbai takes 2 of A, 1 of B and 1 of C and pays Rs. 12 lakh. A buyer in Chennai takes 1, 2 and 1 and pays Rs. 13 lakh. A buyer in Kolkata takes 1, 1 and 2 and pays Rs. 15 lakh. Find each price.

1. The coefficient rows are (2, 1, 1), (1, 2, 1) and (1, 1, 2). Its determinant is 2 times 3 minus 1 times 1 plus 1 times -1, which equals 4. So D is 4.
2. For x, replace the first column with 12, 13, 15. Expanding gives 36 minus 11 minus 17, so Dx is 8. Then x is 8 divided by 4, which is 2.
3. For y, replace the second column. This gives Dy equal to 12, so y is 3.
4. For z, replace the third column. This gives Dz equal to 20, so z is 5.

Machine A costs Rs. 2 lakh, B costs Rs. 3 lakh and C costs Rs. 5 lakh. Check the Mumbai order: 2 times 2 plus 3 plus 5 equals 12. It matches.

Properties of Transpose and Inverse

A few rules cut down your work. The transpose of the transpose returns the original matrix. The transpose of AB is B transpose times A transpose. The inverse of AB is B inverse times A inverse, again with the order reversed. The determinant of A inverse is one divided by the determinant of A.

Two more identities are useful for JEE. For an n x n matrix, A times adj(A) equals |A| times I. As a result, the determinant of adj(A) is |A| raised to the power n minus 1. Students who remember these can answer many multiple-choice questions without any expansion.

To test your formulas on bigger numbers, try the matrix calculator and compare its determinant and inverse with your own. If your inverse has fractions, a fraction calculator can help simplify each entry.

Benefits of Knowing These Formulas

Formulas turn long problems into short ones. With the product rule for determinants, you can find |AB| without multiplying the matrices at all. With the adjoint identity, you can find |adj(A)| in one step. In a timed paper like JEE Main, this saves minutes that you can spend on harder questions.

• Faster solving in board exams and entrance tests.
• Fewer arithmetic mistakes because there are fewer steps.
• A clear base for GATE topics such as eigenvalues and rank.
• Better understanding of how engineering software works behind the screen.

Challenges and Limitations

The formulas look simple on 2 x 2 matrices but grow heavy on larger ones. Finding the adjoint of a 3 x 3 matrix means nine cofactors, each needing its own sign and minor. Cramer’s rule is elegant, yet it becomes slow for four or more variables, so engineers prefer row reduction for large systems.

Another limit is that these formulas only apply to square matrices when you need a determinant or inverse. Rectangular matrices need other tools. Students should also remember that a formula gives an exact answer only if the input numbers are exact.

Common Mistakes to Avoid

1. Dropping the sign pattern. The cofactor sign is (-1) raised to i plus j, so the corners are plus and the middle edges are minus.
2. Forgetting to transpose the cofactor matrix. The adjoint is the transpose, not the cofactor matrix itself.
3. Writing |kA| as k times |A|. For an n x n matrix it is k raised to n times |A|.
4. Using Cramer’s rule when D is zero. Division by zero is not allowed, so choose another method.
5. Not reversing order for inverse of a product. The inverse of AB is B inverse times A inverse.
6. Mixing up Dx and D. Replace only one column of the coefficient matrix at a time.

Best Practices for Using Formulas

1. Write the formula first, then substitute numbers, so that marks are safe.
2. Look for zeros and repeated rows before expanding a determinant.
3. Check that the determinant is not zero before finding an inverse.
4. Verify by multiplying A with your A inverse and looking for the identity.
5. Make a one-page formula sheet and revise it every week.
6. Cross-check with an online tool after you finish your own working.

Conclusion

Matrix formulas are few in number, and they all connect through the determinant. Learn the 2 x 2 versions first, extend them to 3 x 3, and always test your answer. The Indian examples above, from kirana prices to machine sales, show that these formulas solve real money problems, not just textbook sums. Keep practising and the chapter will feel light.

FAQs

What is the formula for the inverse of a matrix?

The inverse of A equals the adjoint of A divided by the determinant of A. It exists only when the determinant is not zero. The adjoint is the transpose of the cofactor matrix.

What is the determinant formula for a 2 x 2 matrix?

For rows (a, b) and (c, d), the determinant is ad minus bc. A result of zero means the matrix is singular.

When can I use Cramer’s rule?

Use it when the number of equations equals the number of unknowns and the main determinant D is not zero. Each variable is a ratio of two determinants.

Is |AB| equal to |A| times |B|?

Yes, this is true for square matrices of the same order. It lets you find the determinant of a product without multiplying the matrices.

Which formulas are most asked in JEE Main?

Determinant properties, the adjoint identity, inverse of a product and Cramer’s rule appear most often. NCERT exercises and previous-year papers cover all of them.