Linear Algebra: Matrices, Determinants & Inverses

It has m rows and n columns.

If A is m \times n and B is n \times p , then AB is m \times p . The inner dimensions must match.

No. In general AB \neq BA , and one may be defined when the other is not.

\begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}

ad - bc

-14

\dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Exactly when \det A \neq 0 . A matrix with determinant 0 is called singular.

a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})

The product of the diagonal entries.

Swapping two rows: multiplies det by -1 Multiplying a row by k: multiplies det by k Adding a multiple of one row to another: no change

\det(AB) = \det A \cdot \det B \det(A^T) = \det A \det(A^{-1}) = \dfrac{1}{\det A} \det(kA) = k^n \det A

B^T A^T and B^{-1}A^{-1}

The square matrix I with 1s on the diagonal and 0s elsewhere, so AI = IA = A

A square matrix equal to its transpose, A = A^T

Each nonzero row has a leading 1, each leading 1 is to the right of the one above, all entries above and below a leading 1 are 0, and zero rows are at the bottom.

The number of pivots in its echelon form, which equals the dimension of its column space (and of its row space).

For an m \times n matrix A : \text{rank}(A) + \text{nullity}(A) = n (the number of columns).

Exactly one, infinitely many, or none. Never exactly two.

\det A \neq 0 Ax = 0 has only the trivial solution Ax = b has a unique solution for every b The columns are linearly independent \text{rank}(A) = n (A row reduces to I )

When the only solution to c_1 v_1 + c_2 v_2 + \cdots + c_k v_k = 0 is c_1 = c_2 = \cdots = c_k = 0 . No vector is a combination of the others.

u \cdot v = u_1 v_1 + \cdots + u_n v_n = \|u\|\|v\|\cos\theta . They are orthogonal when u \cdot v = 0 .