SECTION 2.4 Matrix Algebra

Matrix Arithmetic

For example, A = , is a 3 x 2 matrix. To represent the elements inside the matrix, we may use the notation .  In this case, we can see that a₁₁ = 2,  a₁₂ = 1,  a₂₁ = 3 and so on. Therefore, for an m x n matrix, we will have the notation .  

For each i = 1, 2, ...., m, the ith row of A is the 1 x n matrix and for each j = 1, 2, ...., n, the jth column of A is the m x 1 matrix

Basic algebra of matrices:

1. Equal matrices : Two m x n matrices A = [a_ij] and B = [b_ij] are equal if a_ij = b_ij for each i = 1, 2, ...., m and each j = 1, 2, ..., n.
2. Scalar product of k and A : If k  R and A is matrix, the scalar product of k and A is the matrix obtained from A by multiplying each entry of A by k. It is denoted kA.
3. Sum of two matrices: A = [a_ij] and B = [b_ij] are two m x n matrices. The sum of A and B, denoted by A + B, is the m x n matrix C = [c_ij] such that c_ij = a_ij + b_ij for each i = 1, 2, ...., m and each j = 1, 2, ..., n.
4. Product of two matrices: Let A = [a_ij] be an m x n matrix and B = [b_ij] be an n x p matrix. The product of A and B, denoted by AB, is the m x p matrix [c_ij], where c_ij = a_ikb_kj   for each i = 1, 2, ...., m and each j = 1, 2, ..., p and each k = 1, 2, ..., n._.

Other properties of matrices:

1. Let A be an m x n matrix, B an n x p matrix, and C a p x q matrix. Then (AB)C = A(BC).
   
   
2. If A, B, and C are square matrices of the same size, then A(B+C) = AB+AC and (A+B)C = AC+BC.
   
   
3. If A and B are square matrices of the same size and x and y are real numbers, then

a. x(yA) = (xy)A

b. (x + y)A = xA + yA

c. x(A + B) = xA + xB

d. x(AB) = (xA)B = A(xB)

Diagonals

If A = , an n x n matrix, the main diagonal of A is the set consisting of a₁₁, a₂₂, ...., a_nn. The matrix A is called a diagonal matrix if a_ij = 0 whenever ij.

Consider the following examples:

Let A = .   Then the main diagonal of A is {2, 5, -9}.

Let B = .  Then B is a diagonal matrix.

Other properties:

1. If A and B are diagonal matrices, then the sum of A and B and the product AB are diagonal matrices.

2. If A = , B = , and AB = are n x n matrices, then c_ii = a_iib_ii for each i = 1, 2, ...., n.

The identity matrix of order n is the n x n matrix I_n = [a_ij], where a_ij = 1 if i = j and a_ij = 0 if i  j.

The identity matrix of order 3 is  and the identity matrix of order 4 is

.

Other properties:

If A is an m x n matrix, then I_mA = AI_n = A.

Let A = is an m x n matrix, if there is a n x m matrix B = where b_ij = a_ji for each i = 1, 2, ..., n and each j = 1, 2, ..., m, then B is called the transpose of A, and is denoted by A^t.

Other properties:

1. A square matrix is called symmetric if A = A^t.

2. A matrix all of whose entries are either 0 or 1 is called a zero-one matrix.

Inverse of Matrices

Let A be an n x n matrix. If there exists an n x n matrix B such that AB = BA = I_n, then A is said to be invertible and B is called an inverse of A, denoted by A^-1.

Consider the following example:

Let A = , then B = is the inverse of A, since AB = and BA = .

Other properties:

1. If B and C are inverses of an n x n matrix A, then B = C.

2. If A and B are invertible n x n matrices, then AB is invertible and (AB)^-1 = B^-1A^-1.

3. If A is an invertible n x n matrix, then

a. A^-1 is invertible and (A^-1)^-1 = A.

b. For each n N, Aⁿ is invertible and (Aⁿ)^-1 = (A^-1)ⁿ.

c. For each nonzero k R, kA is invertible and (kA)^-1 = (1/k)A^-1.

4. If A = and ad - bc 0, then A^-1 = 1/(ad-bc) .

Determinants

Let A = be a 2 x 2 matrix. The determinant of A, denoted det(A), is the real number a₁₁a₂₂ - a₂₁a₂₂.

If A = is a 3 x 3 matrix. The determinant of A is the real number a₁₁det() - a₁₂det() + a₁₃det ()

Let n N and let A = be a n x n matrix. The (n - 1) x (n - 1) matrix that is obtained by deleting the rth row and sth column of A is called a minor matrix of A and is denoted by M_rs.

Consider the following example:

Let A = , the minor matrices M₁₄, M₂₂, and M₅₃ of A are

M₁₄ = = (Deleting the first row and fourth column of A),

M₂₂ == (Deleting the second row and second column of A),

and M₅₃ = = .(Deleting the fifth row and third column of A).

Consider A to be a 3 x 3 matrix. By using the minor matrix of A, the determinant of A is a₁₁det(M₁₁) - a₁₂det(M₁₂) + a₁₃det(M₁₃)

Let A = be an n x n matrix. The minor of a_ij is the determinant of M_ij. The number (-1)^i+jdet(M_ij) is called the cofactor of a_ij and is denoted by C_ij.

Let A = be an n x n matrix (n > 3). The determinant of A, denoted by det(A), is a_1k C_1k.

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