medium difficulty8 min

Determinant of a Matrix

The determinant of a square matrix is a scalar value denoted by |A| or det(A). It is defined for square matrices only and can be evaluated by direct formula for order 1 and 2, and by expansion along any row or column for order 3.

In one line: For 2 by 2 determinants: down product minus up product.

Practice This Concept

Understand it

For a 1 by 1 matrix [a], the determinant is a.
For a 2 by 2 matrix, multiply the main diagonal entries and subtract the product of the other diagonal entries.
For a 3 by 3 matrix, choose a row or column, multiply each entry by its cofactor, and add the signed terms.
The value remains the same whichever row or column is chosen, though some choices make calculation shorter.

Example

For A = [[2,3],[5,7]], |A| = 2(7) - 3(5) = 14 - 15 = -1.

Formula and rule

For [[a,b],[c,d]], determinant = ad - bc.

For a 3 by 3 determinant, |A| = a11A11 + a12A12 + a13A13 when expanding along the first row, where Aij is the cofactor of aij.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Evaluate |A| for A = [[1,2,3],[0,4,5],[0,0,6]].

1Expanding along the first column: |A| = 1×|[[4,5],[0,6]]| - 0 + 0 = 1(4×6 - 5×0) = 24.
2Therefore, det(A) = 24.

Method you can reuse

1Check that the matrix is square.
2For order 2, apply ad - bc.
3For order 3, choose a convenient row or column.
4Write each cofactor with its sign.
5Simplify carefully and state the determinant value.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Find the determinant of [[4, -1], [2, 3]].

Think first, then show answer

|A| = 4(3) - (-1)(2) = 12 + 2 = 14.

Common wrong answer

Writing ad + bc instead of ad - bc for a 2 by 2 determinant.

Don't mix this up

Using determinant notation for a non-square matrix. A determinant is not defined for a rectangular matrix.

Exam tip

For a 3 by 3 determinant, expand along the row or column containing the most zeros or simplest numbers.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to evaluate a determinant directly, simplify a determinant containing variables, or identify whether a determinant is zero.

1
Enough for 1 mark
The determinant of a square matrix is a scalar value denoted by |A| or det(A). It is defined for square matrices only and can be evaluated by direct formula for order 1 and 2, and by expansion along any row or column for order 3.
2
Add for 2 marks
Example: For A = [[2,3],[5,7]], |A| = 2(7) - 3(5) = 14 - 15 = -1.
3
Add for 3 marks
For a 1 by 1 matrix [a], the determinant is a.

MCQ Quiz

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