Linear Algebra Essentials

November 14, 2025

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This reference collects identities for inverses, determinants, and traces, followed by common matrix decompositions. Before using an identity, check the matrix dimensions and any requirements for invertibility or symmetry. An identity may fail when these conditions do not hold.

Key Matrix Properties

1. Inverse Properties

These identities apply to invertible square matrices of compatible sizes; the scalar $\kappa$ must be nonzero. In particular, notice the reversed order of the factors in the product formulas. The power identity holds for integer powers.

Property	Formula
Inverse of an Inverse	$(A^{-1})^{-1} = A$
Inverse of a Transpose	$(A^\top)^{-1} = (A^{-1})^\top$
Inverse of a Scalar Multiple	$(\kappa A)^{-1} = \kappa^{-1}A^{-1}$
Inverse of a Product (Two Matrices)	$(AB)^{-1} = B^{-1}A^{-1}$
Inverse of a Product (Three Matrices)	$(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$
Inverse of a Sequence	$(A_1A_2\cdots A_n)^{-1} = A_n^{-1}\cdots A_2^{-1}A_1^{-1}$
Inverse of a Power	$(A^n)^{-1} = (A^{-1})^n$

2. Determinant Properties

Determinants are defined for square matrices. In the product identity, the factors have the same size; in the inverse identity, the matrix must be invertible.

Property	Formula
Determinant of a Transpose	$\det(A^\top) = \det(A)$
Determinant of a Product	$\det(AB) = \det(A)\det(B)$
Determinant of an Inverse	$\det(A^{-1}) = 1/\det(A)$
Determinant of a Scalar Multiple	$\det(\kappa A) = \kappa^n\det(A)$ for an $n\times n$ matrix
Determinant of a Block Diagonal Matrix	$\det(\operatorname{diag}(A,B)) = \det(A)\det(B)$
Determinant of an Identity Matrix	$\det(I) = 1$

3. Trace Properties

The trace is the sum of a square matrix's diagonal entries. Factors in a product may be rectangular, provided each displayed product is defined and square. The cyclic rule lets us move the first factor to the end; it does not let us reorder factors freely.

Property	Formula
Trace of a Transpose	$\operatorname{tr}(A^\top) = \operatorname{tr}(A)$
Trace of a Sum	$\operatorname{tr}(A+B) = \operatorname{tr}(A)+\operatorname{tr}(B)$
Trace of a Product	$\operatorname{tr}(AB) = \operatorname{tr}(BA)$
Trace of a Scalar Multiple	$\operatorname{tr}(\kappa A) = \kappa\operatorname{tr}(A)$
Cyclic Property of Trace	$\operatorname{tr}(ABC) = \operatorname{tr}(BCA) = \operatorname{tr}(CAB)$
Trace of an Identity Matrix	$\operatorname{tr}(I_n) = n$ for the $n\times n$ identity

4. General Matrix Properties

The matrices below are real. Invertibility and eigenvalue decompositions refer to square matrices, while the singular value decomposition also applies to rectangular matrices.

Property	Description
Transpose of a Product	$(AB)^\top = B^\top A^\top$
Rank of a Matrix	Maximum number of linearly independent column vectors in the matrix.
Trace & Determinant for 2×2	For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ , $\operatorname{tr}(A)=a+d$ and $\det(A)=ad-bc$ .
Matrix Invertibility	$A$ is invertible if and only if $\det(A)\ne0$ .
Spectral (Eigen) Decomposition	$A=PDP^{-1}$ when $A$ is diagonalizable, with the eigenvectors in $P$ and eigenvalues in diagonal $D$ . For a real symmetric matrix, $P$ can be chosen orthogonal, so $P^{-1}=P^\top$ .
Singular Value Decomposition (SVD)	$A=U\Sigma V^\top$ , where $U$ and $V$ are orthogonal and the diagonal entries of the rectangular matrix $\Sigma$ are nonnegative singular values.
Orthogonal Matrices	A real square matrix $Q$ is orthogonal if $Q^\top Q=I$ .

Conclusion

The tables separate three common tasks: manipulating inverses, evaluating determinants, and simplifying traces. When applying an identity, first check that the matrix dimensions and required conditions match the problem. The references below develop these properties alongside the decompositions listed in the final table.