Algebra of Matrices: A Quick Review

Algebra of Matrices: Quick Review

• A matrix is a rectangular array of complex numbers arranged in horizontal (rows) and vertical lines (columns).
• Elements or entries are terms for the numbers inside a matrix.
• A matrix with m rows and n columns has an order/type/size of m × n.
• An m × n matrix A is written as A = (aij)m×n, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
• If the number of rows equals the number of columns, then matrix A is a square matrix.
• An n × n square matrix is a square matrix of type n.
• For a square matrix, the principal diagonal is the diagonal from the first element of the first row to the last element of the last row.
• The trace of a square matrix A, denoted tra A, is the sum of the elements on its principal diagonal.
• A matrix is a rectangular matrix if the number of rows differs from the number of columns.
• A zero matrix has every element equal to zero and is denoted by Om×n or O.
• A matrix with just one row is a row matrix.
• A matrix with only one column is a column matrix.
• A square matrix A = (aij)n×n is an upper triangular matrix if aij = 0 whenever i > j.
• A square matrix A = (aij)n×n is a lower triangular matrix if aij = 0 whenever i < j.
• A triangular matrix is a square matrix that is either an upper or a lower triangular matrix.
• A diagonal matrix is a square matrix that is both upper and lower triangular, with all elements equal to 0, except those on the principal diagonal.
• A scalar matrix is a diagonal matrix where all elements on the principal diagonal are equal.
• A unit matrix is a diagonal matrix where every element on the principal diagonal equals unity/one denoted by In.
  • In = (xij)n×n where xij = 1 if i = j, and xij = 0 if i≠ j.
• Two matrices, A and B, are equal if they are of the same type and their corresponding elements are equal, denoted by A = B.

Matrix Sum, Scalar Multiplication, and Conformability

• The sum of two matrices A and B of the same type is a matrix formed by adding their corresponding elements, denoted A + B .
  • Given A = (aij)m×n and B = (bij)m×n, A + B = (cij)m×n where cij = aij + bij.
• Matrix addition is commutative: A + B = B + A, given A and B are of the same type.
• Matrix addition is associative: (A + B) + C = A + (B + C), given A, B, and C are of the same type.
• Matrix O is an additive identity matrix: A + O = O + A = A, for an m × n matrix A.
• For any m × n matrix A, there exists an m × n matrix B such that A + B = B + A = O.
• Matrix B is an additive inverse matrix of A, denoted –A, if A + B = B + A = O.
• The difference of B in A is A + (–B), denoted A – B, given A and B are of the same type.
• Multiplying a matrix A of type m × n by scalar complex number k results in a matrix kA, where every element of A is multiplied by k.
• If A, B are two matrices of same type and k, l are complex numbers:
  • k (A + b) = kA + kB
  • (k + l)A = kA + lA
  • (k l) A = k(lA)
  • 0 A = O
  • kO = O.
• Two matrices, A and B, are conformable for multiplication if the number of columns in A matches the number of rows in B.
  • For A = (aij)m×n and B = (bjk)n × p, their product AB is a matrix C = (cik)m×p, where cij = Σ (aijbjk) from j=1 to n.
• The existence of product AB does not guarantee product BA will also exist.
• Matrix multiplication is not generally commutative; even if AB and BA exist, they may not be equal.
• AB=O does not necessitate that either A or B equals O.
• AB = AC does not necessitate that B = C, even if A ≠ O.
• Matrix multiplication is associative: (AB)C = A(BC), provided conformability is assured.
• Matrix multiplication is distributive over matrix addition, as A(B +C) = AB + AC and (B + C)A = BA + CA when conformability conditions are met.
• For a matrix A of type m × n, AIn = ImA = A.
• If A is a square matrix, then AI = IA = A where I is the multiplicative identity matrix.
• For a square matrix A, AmAn = Am+n = AnAm.
• A square matrix A is an idempotent matrix if A2 = A.
• A square matrix A is an involutory matrix if A2 = I.
• A square matrix A is a nipotent matrix if there exists a positive integer n such that An = O.
  • If n is least positive value, such that An = O, then n is the index of the nilpotent matrix A.
• A transpose of a matrix A, denoted A T or A', is obtained by interchanging the rows and columns.
  • If A = (aij)m×n, then A T = (a'ji)m×n where a'ji = aij.

Transpose, Conjugate, and Special Matrices

• For any matrix A, (AT )T=A.
• For matrices A and B of the same type, (A + B)T = AT + BT.
• For any matrix A and complex number k, (kA)T = kAT.
• If A and B are two matrices for which conformability for multiplication is assured, then (AB)T = BT AT.
• For conformable matrices A1, A2, ..., An, (A1, A2 ...An)T = ATn ATn-1… AT1.
• A square matrix A is symmetric if AT = A.
• A square matrix A is skew-symmetric if AT = –A.
• Every square matrix can be uniquely expressed as the sum of a symmetric and skew-symmetric matrix.
• If A and B are two square matrices of same type then
  • tra (A + B) = tra A + tra B
  • tra (A - B) = tra A – tra B
  • tra (AB) = tra BA
  • tra (kA) = k tra A where k is a complex number.
• A conjugate of matrix A is obtained by replacing each element with its conjugate complex number, denoted A.
• A transposed conjugate is the transpose of the conjugate of matrix A, denoted A  or A.
• A hermitian matrix is a square matrix A that equals its transposed conjugate: A  = A.
• A skew-hermitian matrix is a square matrix A that is the negative of its transposed conjugate: A  = –A.

Inverse Matrix

• The adjoint matrix of a square matrix A, denoted Adj A or adj A, is the transpose of the matrix of cofactors of A where Aij is used to represent corresponding cofactors
  • If A is a square matrix A, then the adjoint matrix of A is: a11 a12 a13  A11 A21 A31  A = a21 a22 a23  Adj A = A12 A22 A32      a31 a32 a33  A13 A23 A33 
• An invertible matrix A is a square matrix for which there exists a matrix B such that AB = BA = I, where I is the identity matrix
• Every squared matrix need not be invertible.
• An invertible matrix has unique inverse.
• The inverse of an invertible matrix A is denoted by A–1, where AA–1 = A–1A = I.
• For a unit matrix I, the inverse exists and I–1 = I.
• If A is invertible, then A–1 is also invertible, and (A–1)–1 = A.
• If matrices A and B are invertible and of the same type, then their product AB is invertible with an inverse that is defined as: (AB)–1 = B–1A–1.
• If any matrix, A, is invertible and k is a nonzero complex number, then kA is invertible where the inverse is defined as: (kA)–1 = k–1A–1.
• If a matrix A is invertible, then its transpose AT is invertible and (AT)–1 = (A–1)T.
• If A is non-singular matrix then A is invertible and A–1 = Adj A/det A.
• For a square matrix A, A(Adj A) = (det A)I.
• If A is an invertible matrix, the determinant of A’s inverse is the inverse of the determinant: det(A–1) = 1/det A.
• For a 2x2 matrix A, where A = [a b; c d], the inverse can be found using: A–1 = 1/(ad − bc) * [d −b; −c a].
• If A, B, and C are square matrices of the same type and A is non-singular, then
  • AB = AC  B = C
  • BA = CA  B = C.
• If A and B are non-singular matrices of same type then Adj(AB) = (Adj B) (Adj A).
• The determinant of the adjoint of a type n square matrix equals the determinant of A to the power of n-1: det (Adj A) = (det A)n−1.
• For 2x2 matrix A, if A = [a b; c d] then Adj A = [d -b; -c a].
• For a 2x2 matrix: 1/a 0 0 A–1 =  0 1/b 0    0 0 1/c

Determinants

• For a 2x2 matrix:
  • A= [a b; c d]
  • Then the determinant of A is ad – bc, denoted as det A or |A| .
• Given element aij, which is in the ith row and jth column of a square matrix A:
  • The cofactor of aij is (–1)i+j multiplied by the minor of aij called cofactor of aij, denoted Aij.

If A has three rows and three columns, then: - a1 b1 c1 - A = a2 b2 c2 - a1A1 + b1B1 + c1C1 = a2A2 + b2B2 + c2C2 = a3A3 + b3B3 + c3C3 = a1A1 + a2A2+a3A3 = b1B1+b2B2+b3B3 = c1C1 + c2C2 + c3C3.

• A determinant of a square matrix A equals the sum of the products of the elements in the first row with their cofactors.
• The determinant of a square matrix A is equal to the sum of the products of a row or column of A with their cofactors.
• A square matrix is singular if its determinant is 0.
• A square matrix is non-singular if its determinant is not 0.
• For a square matrix A, the determinant of A is equal to the determinant of AT: det A = det AT.
• Interchanging any two rows or two columns in a square matrix changes the sign of the determinant.
• Identical rows or columns in a square matrix result in a determinant of zero.
• Multiplying all elements of a row/column in a square matrix by k multiplies the determinant by k.
• If the elements of one row (or column) are respective multiples to the elements of another, the value of the determinant is zero.

Further Properties of Determinants

• If each element in a row/column of a square matrix is the sum of two numbers, then the determinant can be expressed as the sum of the determinants of two square matrices.
• If the elements of a row/column of a square matrix are added to elements of another row/column multiplied by k, then this leaves the value of the determinant is the new determinant.
• The sum of the products of the elements of any row/column of a square matrix coupled with the cofactors of the elements of another row/column is 0.
• The determinant of two square matrices multiplied can be found by multiplying the determinants together: det(AB) = det A * det B.
• The determinant of triangular matrix is obtained by multiplying components of principle diagonal.
• An example determinant formula, with values a, h, g, b, f, c, is:
  • a h g
  • h b f = abc + 2fgh − af 2 − bg2 − ch2
  • g f c
• An example determinant formula, with values a, b, c, is:
  • a b c
  • b c a = 3abc − a3 − b3 − c3
  • c a b
• The determinant:
  • 1+ a b c
  • a 1+ b c = 1+ a+ b+ c
  • a b 1+ c
• The determinant:
  • 1 a a2 1 a bc
  • 1 b b2 = 1 b ca = (a − b)(b − c)(c − a)
  • 1 c c2 1 c ab
• The determinant:
  • 1 a a3 1 a2 bc
  • 1 b b3 = 1 b2 ca = (a − b)(b − c)(c − a)(a + b + c)
  • 1 c c3 1 c2 ab
• The determinant:
  • 1 a2 a3 a a2 bc
  • 1 b2 b3 = b b2 ca = (a − b)(b − c)(c − a)(ab + bc + ca)
  • 1 c2 c3 c c2 ab

Linear Equations

• A system of simultaneous linear equations in two variables is such that a1x+b1y = c1, a2x + b2y= c2
• If A= [a1 b1; a2 b2], X= [x; y], and B= [c1; c2], then the system can be written as a matrix equation AX = B.
• If x =  and y =  such that ,  are complex numbers, satisfies the equations of system so x = , y =  called solution of system (1).
• The system: a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z =d3 can be expressed with matrices.
  • Thus If A = [a1 b1 c1; a2 b2 c2; a3 b3 c3 ], X = [x; y; z], and B = [d1 d2 d3], the system can be written as AX=B.
• The system of equations AX = B is consistent if a solution exists; otherwise, it is inconsistent.
• When A is a non-singular matrix, A −1/B is the solution of AX = B, and this solution is unique.
• In Cramer's Rule:
  • a1 * 1+ b1* 2 + c1* 3= d1, a2 * 1+ b2* 2 + c2* 3 = d2, a3* 1+ b3* 2 + c3* 3 =d3
  • The determinant of the system is:  = [a1 b1 c1; a2 b2 c2; a3 b3 c3].
  • 1 = [d1 b1 c1; d2 b2 c2; d3 b3 c3], 2 = [a1 d1 c1; a2 d2 c2; a3 d3 c3].
  • 3 = [a1 b1 d1; a2 b2 d2 a3 b3 d3] then x =1/, y= 2/ ,z= 3/
• Coefficient Matrix can be created, such that: a1 b1 c1; a2 b2 c2; a3 b3 c3
• Augmented matrix can be created from any matric equation and is such that: a1 b1 c1 d1; a2 b2 c2 d2; a3 b3 c3 d3]

Elementary Transformations and Rank

• Elementary transformations are operations on rows/columns: interchange, scalar multiplication, or adding a multiple of one row/column to another.
  • Ri  Rj denotes the interchange of the ith and jth rows,
  • Ri → kRi or simply kRi denotes scalar multiplication of the ith row while the addition of multiple rows is denoted as Ri → Ri + kRj or simply Ri + kRj.
• The Gauss-Jordan method can be used to solve linear systems, which reduces the augmented matrix using elementary row transformations to 1 0 0 ; 0 1 0 ; 0 0 1  Then the solution equals [ , , and  ].
• Submatrix: A matrix obtained by erasing some rows or columns from and existing matrix.
• If B can be made form A (is a square submatrix of order r), then det B is know as an r-rowed miner of A
• To use the Rank Theorems:
• Non-zero is non-zero matrix: A positive integer r is said to be of r of if
• (i) there is a non-zero A of of r, where A also has r-rowed minor.
• (ii) every (r + 1)-rowed minor of A (if exists) is zero.
• If there is a zero, then the number equals 0, zero = matrix that must have Rank = 0
• Important Matrix info: (the amount of rank)
• non-zero 3 × 3 can equal its rank= 3. Then If A, can get almost 2 x 2 and rank= 2. When every 2 × 2 does = 1.
• The above rules do not influence the rank of a matric.
• Rank does not impact on chains that are produced from this.
• Systems can be consistent. Can be found from a coefficient. Has to be same with aug.
• If you have unknown and rank then you need these to figure this out/ Then aug Matric must come from these.
• Case 1: if first is ! the second then the equation has no answer and inconsistent.
• Case 2: the first has to correspond. If it doesn't have a equal value, must find the solution.
• The system of first or second equal and < n. The system can be consistently many solutions.
• The matrix equations. When the value = 0. They equal homogeneous in x, y, z.
• All equal 0. Always for any solver of a set of liner equations.
• AX = has to use operation on augmatric A . Can make from AX =B to reduced form. p 1 p2 p3 p4 
• 0 p5 p6 p7 . To make work you need to use operation form. By reducing. 0 0 p8 p9
• To fine what its =
• To use this. If p8  0, then singular and can solve matrix A = B = has 1 solution.
• With p8 = 0, p9  0. A solution does exsisterMatrix A = B = and has = (0).
  • This means it can not be solved.
• As if A, is a real world model. It has to have one or many can be an answer.
• As if P8/ p9 = has = (0)= a lot can make from matrix equation A= B = many real world solutions.
• If p = 0 then no answer exist. Then is the unit matrix can do something and see result= to get n.
• For the non. Is = to rank which the solution will take the number needed.