Adjoint and inverse of a matrix

What is the adjoint of a matrix?

In plain words: replace every element of A by its cofactor, then turn the rows into columns. For a 3 × 3 matrix, the cofactors of row 1 become column 1 of adj A:

The 2 × 2 shortcut. For a matrix of order 2, the cofactors are signed single entries, and adj A comes from swapping the two diagonal elements and changing the signs of the other two (NCERT’s Remark, page 88). NCERT Example 12: for A with rows 2, 3 and 1, 4, the cofactors are A₁₁ = 4, A₁₂ = −1, A₂₁ = −3 and A₂₂ = 2, so adj A has rows 4, −3 and −1, 2.

A(adj A) = |A| I

NCERT Theorem 1 (page 88): for any square matrix A of order n,

Why: the (i, j)th element of A(adj A) is row i of A times column j of adj A, and column j of adj A holds the cofactors of row j of A. When i = j, this is a row times its own cofactors, which is |A|. When i ≠ j, it is a row times another row’s cofactors, which is 0. So every diagonal element is |A| and every other element is 0.

For the 2 × 2 matrix with rows a, b and c, d, A(adj A) has rows ad − bc, −ab + ba and cd − dc, −cb + da, that is (ad − bc)I.

Singular matrices and |AB| = |A||B|

NCERT’s examples (page 89): the matrix with rows 1, 2 and 4, 8 has determinant 8 − 8 = 0, so it is singular; the matrix with rows 1, 2 and 3, 4 has determinant 4 − 6 = −2 ≠ 0, so it is non-singular.

The determinant of the product of matrices is equal to product of their respective determinants

NCERT Mathematics Part I (Class 12), Section 4.5, page 89

That is, |AB| = |A||B| for square matrices of the same order (NCERT Theorem 3, stated without proof). So if A and B are non-singular, so are AB and BA (NCERT Theorem 2). Taking determinants of (adj A)A = |A| I for a 3 × 3 matrix gives |adj A||A| = |A|³, so |adj A| = |A|² when |A| ≠ 0 (NCERT’s Remark, pages 89–90). In general, for a matrix of order n ≥ 2 (including singular matrices):

For a singular matrix, cancellation by |A| is unavailable. The identity still holds because both sides are polynomials in the entries and agree for every nonsingular matrix; one may approach a singular matrix by nonsingular ones.

Beyond the NCERT text (JEE): |adj (adj A)| = |A|^{(n − 1)²}, and since each cofactor of kA is k^{n − 1} times that of A, adj (kA) = k^{n − 1} adj A. For a 3 × 3 matrix with |A| = 3, |adj A| = 9 and |adj (adj A)| = 81. But |A + B| is not |A| + |B|: for A = B = I of order 2, |A + B| = |2I| = 4.

The inverse from the adjoint

A square matrix A is invertible if and only if A is nonsingular matrix.

NCERT Mathematics Part I (Class 12), Section 4.5, page 90

NCERT Theorem 4 (page 90). If A is invertible, AB = I for some B, so |A||B| = |I| = 1 and |A| cannot be 0. Conversely, if |A| ≠ 0, divide A(adj A) = (adj A)A = |A| I by |A|:

As with an invertible function, the inverse undoes A from either side: AA^⁻¹ = A^⁻¹A = I. For a 2 × 2 matrix with rows a, b and c, d and ad − bc ≠ 0, the formula gives 1/(ad − bc) times the matrix with rows d, −b and −c, a. Taking determinants of AA^⁻¹ = I also gives |A^⁻¹| = 1/|A|.

Solved examples

Common mistakes to avoid

Quick revision summary

• adj A is the transpose of the matrix of cofactors; for order 2, swap the diagonal elements and change the signs of the other two.
• A(adj A) = (adj A)A = |A| I, and |adj A| = |A|^ⁿ⁻¹ for a matrix of order n.
• A is invertible exactly when it is non-singular (|A| ≠ 0), and then A^⁻¹ = (1/|A|) adj A; also |AB| = |A||B| and |A^⁻¹| = 1/|A|.

Questions students ask

Is there a shortcut for the adjoint of a 3 × 3 matrix?

Beyond the NCERT text, each cofactor of a 3 × 3 matrix comes out with its sign if rows and columns are taken in cyclic order (1 after 3): for the element in row i, column j, take the two rows after i and the two columns after j, and in that 2 × 2 block subtract top right × bottom left from top left × bottom right. For rows 2, 1, 3; 4, 0, 1 and 1, 5, 2, the element 1 in row 1, column 2 takes rows 2, 3 and columns 3, 1, so A₁₂ = 1 × 1 − 4 × 2 = −7, which is −(4 × 2 − 1 × 1). The cyclic order swaps the two rows only when i = 2 and the two columns only when j = 2, giving the sign (−1)^{i + j}, and it works only for order 3; adj A is their transpose.

What is the difference between the adjoint and the inverse of a matrix?

The adjoint adj A exists for every square matrix, while the inverse A^⁻¹ exists only when |A| ≠ 0, and then A^⁻¹ = (1/|A|) adj A: the inverse is the adjoint divided by the determinant. So the two are equal only when |A| = 1 (and A^⁻¹ = −adj A when |A| = −1), and they behave differently: A(adj A) = |A| I but AA^⁻¹ = I, and |adj A| = |A|^ⁿ⁻¹ but |A^⁻¹| = 1/|A|. For the matrix with rows 3, 1 and 4, 2, |A| = 2, adj A has rows 2, −1 and −4, 3, and A^⁻¹ has rows 1, −1/2 and −2, 3/2.

What is adj (adj A), the adjoint of the adjoint?

Beyond the NCERT text, adj (adj A) = |A|^ⁿ⁻² A for order n ≥ 3, with the order-2 case stated separately as adj (adj A) = A (even when A is singular): for order 2 it is A itself, and for order 3 it is |A| A. When |A| ≠ 0 it follows from the adjoint identity applied to adj A, whose determinant is |A|^ⁿ⁻¹, after multiplying on the left by A and dividing by |A|. For the matrix with rows 1, 1, 0; 0, 1, 1 and 1, 0, 1, |A| = 2, adj A has rows 1, −1, 1; 1, 1, −1 and −1, 1, 1, and its adjoint has rows 2, 2, 0; 0, 2, 2 and 2, 0, 2, which is 2A.

Is the adjoint of a matrix the same as its adjugate or its transpose?

NCERT’s adjoint adj A, the transpose of the matrix of cofactors, is not the transpose A′ of A itself, and beyond the NCERT text many books call it the adjugate (or classical adjoint). The other name avoids a clash, since in university linear algebra “adjoint” usually means the conjugate transpose, which for a real matrix is just A′, so a “self-adjoint” matrix there is a symmetric (or Hermitian) one (Wikipedia, “Adjugate matrix” and “Hermitian matrix”). For the matrix with rows 6, 1 and 4, 3, A′ has rows 6, 4 and 1, 3, while adj A has rows 3, −1 and −4, 6.

How many 2 × 2 matrices with entries 0 or 1 are singular?

Of the 2⁴ = 16 matrices of order 2 with every entry 0 or 1, 10 are singular and 6 are non-singular. The matrix with rows a, b and c, d is singular when ad = bc, and here both products are 0 or 1: ad = bc = 1 only for the matrix of four 1s, while ad = 0 in 3 ways, (a, d) = (0, 0), (0, 1) or (1, 0), and bc = 0 in 3 ways, giving 3 × 3 = 9 more. For order 3, a computer check of every such matrix finds 338 singular ones.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Find the adjoint of the matrix with rows 1, 2 and 3, 4, and of the matrix with rows 1, −1, 2; 2, 3, 5; −2, 0, 1 (NCERT Exercise 4.4, Q1 and Q2).

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   For the 2 × 2 matrix, adj A has rows 4, −2 and −3, 1. For the 3 × 3 matrix, the cofactors row by row are 3, −12, 6; 1, 5, 2; −11, −1, 5, so adj A has rows 3, 1, −11; −12, 5, −1; 6, 2, 5.

2. Q2. Find the inverses of the matrices with rows 2, −2 and 4, 3, and with rows −1, 5 and −3, 2 (NCERT Exercise 4.4, Q5 and Q6).

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   |A| = 6 + 8 = 14, so A^⁻¹ = (1/14) times the matrix with rows 3, 2 and −4, 2. For the second, |A| = −2 + 15 = 13, so A^⁻¹ = (1/13) times the matrix with rows 2, −5 and 3, −1.

3. Q3. Find the inverse of the matrix with rows 1, −1, 2; 0, 2, −3; 3, −2, 4 (NCERT Exercise 4.4, Q10).

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   Along row 1: |A| = 1(8 − 6) + 1(0 + 9) + 2(0 − 6) = 2 + 9 − 12 = −1. adj A has rows 2, 0, −1; −9, −2, 3; −6, −1, 2, so A^⁻¹ = −adj A, with rows −2, 0, 1; 9, 2, −3; 6, 1, −2.

4. Q4. For A with rows 3, 2 and 1, 1, find the numbers a and b such that A² + aA + bI = O (NCERT Exercise 4.4, Q14).

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   A² has rows 11, 8 and 4, 3. Comparing elements: (1, 2) gives 8 + 2a = 0, so a = −4; (1, 1) gives 11 + 3a + b = 0, so b = 1. Check (2, 2): 3 + a + b = 0. So a = −4, b = 1.