Linear Algebra Exam Two

I'm gonna cry a ton T-T

aij

The scalar entry in the ith row and jth column (diagonal entries in an mxn matrix A are a11, a22, a33, … (i=j)

Mian Diagonal

Defined on an mxn matrix

Antidiagonal/Counter Diagonal

Defined only on a square matrix Anxn
aij, (i + j = n +1)

Diagonal Matrix

A square (n x n) matrix whose non diagonal entries are zero (ex = identity matrix In)

Zero Matrix

All entries are zero

When are two matrices equal?

If they have the same size and their corresponding entries are zero

Matrix Sums

If A and B are mxn matrices (same size), each entry in A + B is the sum of the corresponding entries

Scalar Multiples

If A is an mxn matrix and r is a scalar, then any entry of rA is r times the corresponding entry in A

Theorem 2.1

Let A, B, and C are matrices of the same size and let r and s be scalars

• A + B = B + A

• (A + B) + C = A + (B + C) = (A + C) +B

• A + 0 = A

• r(A + B) = rA + rB

• (r + s)A = rA =sA

  • r(sA) = (rs)A = s(rA)

Matrix Multiplication

Bx = x1b1 + x1b1 + x2b2 + … + xnbn

If A is an mxn matrix, B is an nxp matrix with columns [b1 b2 … bp], then AB is the m x p matrix whose columns are Ab1, …, Abp

Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B

Row-Column Rule for Computing AB

If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding entries from row i of A and column j of B

(AB)ij = ai1b1j + ai2b2j + … + ainbnj

Properties of Matrix Multiplication: Theorem 2.2

Let A be an mxn matrix, and let B and C have sides for which the indicated sums and products are defined

• A(BC) = (AB)C

• A(B + C) = AB + AC]

• (B +C)A = BA + CA

• r(AB) = (rA)B = A(rB) for any scalars r

  • InA = A = AIn

Powers of a Matrix

If A is an nxn matrix, and if k is a positive integer, then
A^k = AAA…A (copies of A’s)

Transpose of a Matrix

Given an mxn matrix A, the transpose of A is the nxm matrix, denoted by A^T, whose column are formed from the corresponding dows of A (Aij = (A^T)ji)

Theorem 2.3

• (A^T)^T = A

• (A + B)^T = A^T + B^T

• For any scalar r, (rA)^T = rA^T

• (AB)^T = B^TA^T