Matrices and Determinants

Determinant evaluation

A determinant is the scalar \det A attached to a square matrix A. For a third-order matrix it is computed by cofactor expansion along a chosen row or column: \det A=\sum_{j=1}^{3}(-1)^{i+j}a_{ij}M_{ij} for a fixed row i, where M_{ij} is the minor obtained by deleting row i and column j. The sign (-1)^{i+j} follows a checkerboard pattern with + at position (1,1). In practice, expand along the row or column that contains the most zeros, because each zero removes a term. For example, if the second row is (0,0,k), the expansion reduces to the single term (-1)^{2+3}k\,M_{23}.

Many determinants can be decided without full evaluation. A determinant with a zero row or zero column is zero. The same holds when every entry of one column carries a common factor that vanishes at a particular value of the variable, so substituting that value is a fast test for a root. For a determinant that is a polynomial in x, the constant term e equals the value at x=0, because every other term carries a positive power of x.

Rows or columns that are equal, proportional, or linearly dependent make the determinant zero. Such dependence often hides in the pattern of the entries. When a_{ij} is linear in i and j, as in 2(i+j), the rows of a third-order matrix are dependent and the determinant vanishes. If the entries are logarithms of the terms of a geometric progression, the logarithms form an arithmetic progression, so the rows are dependent and the determinant is zero. A square matrix with a linearly dependent set of columns is singular and has no inverse; for instance, a column equal to the sum of two other columns forces singularity. In trigonometric matrices, one row can be a constant multiple of another for every value of the angle, which makes the matrix singular for any value of the remaining parameter.

Cancellation can also produce zero. When the expansion gives two products that are equal and opposite, the terms cancel completely. In a triangle, B+C=\pi-A gives \tan(B+C)=-\tan A, so a bracket in the expansion vanishes and the determinant is zero. The law of cosines a^2=b^2+c^2-2bc\cos A makes the bracket a^2-b^2-c^2+2bc\cos A vanish, so the determinant is zero. The identities 1-2\sin^2\theta=\cos 2\theta and 2\cos^2\theta-1=\cos 2\theta are often needed to turn entries into proportional rows.

A common factor of one row or column may be taken outside the determinant. Factoring different constants from different columns multiplies the values, so scaling one column by p and another by q multiplies the determinant by pq, not by p+q. For a scalar m and an n\times n matrix, \det(mA)=m^{n}\det A, because each of the n rows contributes one factor m; this follows from the single-row rule applied n times. In particular \det(2I_3)=2^{3}=8, so a matrix that is twice the identity has determinant 8.

When each column is a linear combination of the columns of a known matrix, expand the determinant term by term. Any term in which the same base row appears in two columns vanishes, and only the terms with three distinct base rows survive. The coefficients multiply to give the overall factor, and the order of the base rows fixes the sign. Interchanging two rows or columns changes the sign, while a cyclic reordering of three rows is an even permutation and leaves the value unchanged. Relabelling rows or columns by a permutation multiplies the determinant by the sign of that permutation, and a quick way to fix the sign is to test the identity matrix. If a family of determinants is closed under interchanging two columns, the values of each determinant and its partner are opposite, so the total sum is zero.

Replacing a column by itself minus combinations of other columns leaves the determinant unchanged. For example, the operation C_3\to C_3-C_1-C_2 can simplify a matrix before expansion, and then comparing with a given relation determines an unknown constant. Replacing the first column by the sum of all columns can produce a common factor. If that common factor is 2p, the remaining matrix is triangular, and the determinant follows from the diagonal. For a linear system whose coefficient determinant D is nonzero, Cramer's rule gives x_i=D_i/D, where D_i is obtained by replacing column i with the constant terms.

Some standard forms are worth memorising. The circulant matrix with rows (a,b,c), (b,c,a), (c,a,b) has determinant -(a^3+b^3+c^3-3abc)=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca). Hence it vanishes when a+b+c=0 or when a^3+b^3+c^3=3abc, because the second condition is equivalent to the vanishing of the second factor. The identity x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx) shows that if x+y+z>0, the equality forces x=y=z. Applied to the sines of a triangle, equal sines imply equal sides by the sine rule, and a matrix with equal sides has equal rows and zero determinant.

For two determinants of the form \begin{vmatrix}1&p&q\\1&q&r\\1&r&p\end{vmatrix} and its cyclic companion, the expansion gives -(p^2+q^2+r^2-pq-qr-rp) in each case. Their sum is -[(p-q)^2+(q-r)^2+(r-p)^2], which is negative whenever the quantities are not all equal. A Vandermonde determinant with rows (1,1,1), (a,b,c), (a^2,b^2,c^2) equals (b-a)(c-a)(c-b)=(a-b)(b-c)(c-a). If every column has the form (x,x^2,x^3)^T, first factor abc from the columns, and the remaining determinant is this Vandermonde determinant.

The matrix determinant lemma states \det(I+uv^{T})=1+v^{T}u. For a column vector A of order 3\times 1, \det(I_3+AA')=1+A'A, which is the sum of the squares of the entries plus one. When a determinant is a polynomial in a parameter \alpha such as (b^2-ac)(a\alpha^2+2b\alpha+c), vanishing for every \alpha>0 forces the constant factor to vanish, so b^2=ac and a,b,c are in geometric progression. If b^2\neq ac, the determinant vanishes only at the roots of a\alpha^2+2b\alpha+c=0, and these roots are the values that solve the determinant equation.

For a matrix whose entries are cosines, sines and tangents, expand carefully and simplify with identities; the value is often 0 or a simple trigonometric expression. A determinant with entries in the form of \cos, \tan and \sin can attain a minimum over \theta after reducing to 1-3\sin^2\theta\cos^2\theta, which equals 1-\frac{3}{4}\sin^2 2\theta, with minimum \frac14. When the first column of a matrix is (1,1,1)^T, the expansion gives 1+xy+yz+zx, and for x=a/(b-c), y=b/(c-a), z=c/(a-b) the symmetric sum xy+yz+zx equals -1, so the determinant is zero.

Several entries of a determinant may be given by a formula in an index k. Here \det A_k=2k-1, so the sum over k from 1 to n equals \sum_{k=1}^{n}(2k-1)=n^{2}. A third-order determinant whose entries are all \pm 1 is always divisible by 2^{3-1}=4, so it is even; values 0 and \pm 4 both occur. Finally, a determinant with a parameter in one row may be zero because two of its rows coincide or because the row-scaling identity makes it singular, and each such case is checked by the rules above.

Adjugate matrix properties

The adjugate (classical adjoint) of a square matrix A of order n, written \operatorname{adj}A, is the transpose of the matrix of cofactors of A. Its fundamental identity is A(\operatorname{adj}A) = (\operatorname{adj}A)A = (\det A)\,I_n, which holds for every square matrix, singular or not. Because the product is a scalar multiple of the identity, the scalar \det A must never be dropped. For example, if a 3\times 3 matrix has \det A = 2, then A(\operatorname{adj}A) = 2I, not I.

For a non-singular matrix A, the adjugate satisfies \operatorname{adj}A = (\det A)A^{-1}. It is therefore non-singular, and (\operatorname{adj}A)^{-1} = \dfrac{A}{\det A}. In particular, when \det A = 1, (\operatorname{adj}A)^{-1} = A. Since A and \operatorname{adj}A commute, AB with B = \operatorname{adj}A is the scalar matrix (\det A)I, which is never the null matrix when \det A \neq 0.

For a matrix with A^{2} = I we have A^{-1} = A, so \operatorname{adj}A = (\det A)A. When \det A = -1, this gives \operatorname{adj}A = -A, so A + \operatorname{adj}A and A^{-1} + \operatorname{adj}A are both null matrices.

The following rules are used repeatedly for square matrices of order n:

• \det(\operatorname{adj}A) = (\det A)^{n-1} — for order 3 this is (\det A)^{2}. The determinant of the cofactor matrix C is the same quantity, because C = (\operatorname{adj}A)^{T}.
• \operatorname{adj}(kA) = k^{n-1}\operatorname{adj}A — for order 3, \operatorname{adj}(kA) = k^{2}\operatorname{adj}A, so |\operatorname{adj}(kA)| = k^{6}|A|^{2}. Also |\operatorname{adj}(M)| = |M|^{2} for a 3\times 3 matrix M, so |\operatorname{adj}(3A)| = 729|A|^{2}. For a scalar c, |c\,\operatorname{adj}(kA)| = c^{3}k^{6}|A|^{2} at order 3.
• \operatorname{adj}(A^{T}) = (\operatorname{adj}A)^{T} — the adjugate commutes with transposition, so A(\operatorname{adj}A^{T}) = A(\operatorname{adj}A)^{T} is a valid identity.
• \operatorname{adj}(AB) = (\operatorname{adj}B)(\operatorname{adj}A) — the order of the factors is reversed, so \operatorname{adj}(AB) is in general not equal to (\operatorname{adj}A)(\operatorname{adj}B) or to \operatorname{adj}(BA). However, (AB)\operatorname{adj}(AB) = |AB|I_n always holds.

For a non-singular matrix A, A^{2}=A forces A = I, because multiplying by A^{-1} gives A = I. The weaker relation A^{3} = A only gives A^{2} = I, which allows matrices such as diagonal matrices with entries \pm 1 that are not the identity.

Inverse of a matrix

A square matrix A has an inverse A^{-1}, with AA^{-1} = A^{-1}A = I, exactly when \det A \neq 0. A matrix with zero determinant is singular and has no inverse. For a 2\times 2 matrix \begin{bmatrix}p & q\\ r & s\end{bmatrix} with ps - qr \neq 0, the inverse is \dfrac{1}{ps-qr}\begin{bmatrix}s & -q\\ -r & p\end{bmatrix}. If the inverse B of an unknown matrix is given, then the unknown matrix is A = B^{-1}, which is found by applying this formula to B.

The determinant of an inverse is the reciprocal of the determinant: \det(A^{-1}) = \dfrac{1}{\det A}.

For non-singular matrices A and B of the same order, the inverse of a product reverses the order of the factors: (AB)^{-1} = B^{-1}A^{-1}. The statement (AB)^{-1} = A^{-1}B^{-1} is false in general. Similarly (AB)^{T} = B^{T}A^{T}, so the order also reverses for transposes. Hence (BA)(AB)^{-1} = BAB^{-1}A^{-1}, which equals I only when AB = BA.

Cancellation works only with an invertible factor. If A is non-singular, AB = AC implies B = C, because multiplying on the left by A^{-1} gives B = C. Conversely, BX = CX for every column matrix X implies B = C, since choosing X as each standard basis column shows that every column of B - C is zero.

For non-singular A and B of the same order, AB = A gives A(B - I) = O, so B = I. Then BA = B gives A = I. Thus both relations force A = B = I, and the identities A^{2} = A and AB^{2} = A^{2}B hold.

The product of two non-zero matrices can be the null matrix, so AB = O does not force A = O or B = O. For example, two non-zero matrices with nonzero entries in suitably placed positions can multiply to zero. However, if AB = I for square matrices of the same order, then B = A^{-1}, so BA = I = AB.

If A has \det A = 1 and A^{-1} = A^{T}, then \operatorname{adj}A = \det(A)A^{-1} = A^{T}, so the inverse, the adjugate and the transpose all coincide. A skew-symmetric matrix must have zero diagonal entries, so a matrix whose diagonal entries are nonzero cannot be skew-symmetric.

Symmetric and skew-symmetric matrix decomposition

A square matrix A is symmetric when A^{T} = A and skew-symmetric when A^{T} = -A. Every square matrix can be written uniquely as the sum of a symmetric part and a skew-symmetric part: A = P + Q, where P = \dfrac{1}{2}(A + A^{T}) is symmetric and Q = \dfrac{1}{2}(A - A^{T}) is skew-symmetric. Entrywise, P_{ij} = \tfrac{1}{2}(a_{ij} + a_{ji}) and Q_{ij} = \tfrac{1}{2}(a_{ij} - a_{ji}).