MAT 2401Linear Algebra

2.2 Properties of Matrix Operations

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Today

Written HW

Review

We have defined the following matrix operations

“term-by-term” operations•Matrix Addition and Subtraction

•Scalar Multiplication Non-“term-by-term” operations

•Matrix Multiplication

Review

We have studied some of the properties such as…•AI=IA=A

In general, •AB≠BA

•AB=0 does not imply A=0 or B=0

Preview

Look at more properties about these operations.

Most of the properties are natural to conceive (inherited from the number system).

Sometimes, it may be more effective to remember what properties are not true.

Preview

Most properties come with names. We will not emphasize on them.

Look at another operation: Transpose

Matrix Addition and Scalar Multiplication

Let A,B,C be mxn matrices, 0 the mxn zero matrix, and c and d scalars.

1. A + B = B + A2. (A + B) + C = A + (B + C)3. c(dA) = (cd)A4. c(A + B) = cA + cB5. (c + d)A = cA + dA

Matrix Addition and Scalar Multiplication

Let A,B,C be mxn matrices, 0 the mxn zero matrix, and c and d scalars.

6. A + 0 = A7. A + (-A) = 08. If cA=0, then either c=0 or A=0

Example 1

Solve the matrix equation 3X+A=Bwhere 1 0 0 1

, 1 2 1 2

A B

Matrix Multiplication

Let A,B,C be matrices of the appropriatesizes, I a suitably sized identity matrix, and c and d scalars.

1. (AB)C = A(BC)2. A(B+C)=AB+AC3. (A+B)C = AC+BC4. c(AB)=(cA)B=A(cB)

Cancellation Law

Q: Does AC=BC imply A=B?A:

Matrix Power

Let A be a square matrix, k a non-negative integer.

times

if 0

if 0k

k

I kA A A A k

Laws of Exponents

Let A be a square matrix, i, j, k non-negative integers.

1. AiAj =2. (Ai)j =3. 0k =4. Ik =

Transpose of a Matrix

Let A=[aij] be a mxn matrix, the transpose of A is the nxm matrix AT so that the (i,j)th entry of AT is aji.

(Interchanging the rows and columns of A)

Transpose of a Matrix

Let A=[aij] be a mxn matrix, the transpose of A is the nxm matrix AT so that the (i,j)th entry of AT is aji.

21 22 23 24

31 32 33 34

11 12 13 14

TA a a a a A

a a a a

a a a a

Example 2

1 2

1 0

1 2 3 1

2 2 3 1

[ ]

T

T

T

A A

B B

C x y z C

Scratch:Q: What is the dimension of the transpose?

Properties of Matrix Transpose

Let A,B be matrices of the appropriatesizes, and c a scalar.

1. (AT)T= A2. (A + B)T = AT + BT

3. (cA)T = cAT

4. (AB)T = BTAT

Properties of Matrix Transpose

Let A,B be matrices of the appropriatesizes, and c a scalar.

1. (AT)T= A2. (A + B)T = AT + BT

3. (cA)T = cAT

4. (AB)T = BTAT Why?

Example 3

1 0 0 1,

1 2 1 2

T

T T

A B

AB

AB

B A

Symmetric Matrix

Symmetric

A square matrix is symmetric if aij=aji

for all i,j.

Properties of Symmetric Matrices

1. If A is symmetric, then AT = In fact, A is symmetric if and only if AT =

2. AAT and ATA are symmetric for any matrix A.

Properties of Symmetric Matrices

1. If A is symmetric, then AT = In fact, A is symmetric if and only if AT =

2. AAT and ATA are symmetric for any matrix A..

Why?