Many people have difficulty dealing with matrices, but in fact it’s very easy, All you need to know is the basics in solving matrices.
I wrote this article today and it is a summary of what I know and I hope you will benefit and see how easy it is.
Simple defenition of matrix A rectangular array of numbers is called a matrix.
We shall mostly be concerned with matrices having real numbers as entries.
The horizontal arrays of a matrix are called its rows and the vertical arrays are called its columns. A matrix having m rows and n columns is said to have the order m× n.
A matrix A of order m× n can be represented in the following form:
A matrix having only one column is called a column vector; and a matrix with only one row is called a row vector.
Whenever a vector is used, it should be understood from the context whether it is a row vector or a column vector.
Can a matrix be equal to another matrix?
Two matrices A = [aij ] and B = [bij ] having the same order m× n are equal if aij = bij for each i = 1, 2, . . . ,m and j = 1, 2, . . . , n.
In other words, two matrices are said to be equal if they have the same order and their corresponding
entries are equal.
For example The linear system of equations 2x + 3y = 5 and 3x + 2y = 5 can be identified with the matrix
Special Matrices definitions
1. A matrix in which each entry is zero is called a zero-matrix, denoted by 0. For example and
2. A matrix having the number of rows equal to the number of columns is called a square matrix. Thus, its order is m× m (for some m) and is represented by m
3. In a square matrix, A = [aij ], of order n, the entries a11, a22, . . . , ann are called the diagonal entries and form the principal diagonal of
4. A square matrix A = [aij ] is said to be a diagonal matrix if aij = 0 for i 6= j. In other words, the non-zero entries appear only on the principal diagonal. For example, the zero matrix 0n and are a few diagonal matrices.
A diagonal matrix D of order n with the diagonal entries d1, d2, . . . , dn is denoted by D = diag(d1, . . . , dn).
If di = d for all i = 1, 2, . . . , n then the diagonal matrix D is called a scalar matrix.
5. A square matrix A = [aij ] with aij = is called the identity matrix,
denoted by In.
The subscript n is suppressed in case the order is clear from the context or if no confusion arises.
6. A square matrix A = [aij ] is said to be an upper triangular matrix if aij = 0 for i > j.
A square matrix A = [aij ] is said to be an lower triangular matrix if aij = 0 for i < j.
A square matrix A is said to be triangular if it is an upper or a lower triangular matrix.
Operations on Matrices
- Transpose of a Matrix The transpose of an m × n matrix A = [aij ] is defined as the n ×m matrix B = [bij ], with bij = aji for 1 ≤ i ≤ m and 1 ≤ j ≤ n. The transpose of A is denoted by At.
That is, by the transpose of an m × n matrix A, we mean a matrix of order n × m having the rows of A as its columns and the columns of A as its rows.
Thus, the transpose of a row vector is a column vector and vice-versa.
- Theorem: For any matrix A, we have (At)t =
Proof. Let A = [aij ], At = [bij ] and (At)t = [cij ]. Then, the definition of transpose gives cij = bji = aij for all i, j and the result follows.
- Addition of Matrices
let A = [aij ] and B = [bij ] be are two m×n Then the sum A + B is defined to be the matrix C = [cij ] with cij = aij + bij .
Note that, we define the sum of two matrices only when the order of the two matrices are same.
Multiplying a Scalar to a Matrix
Let A = [aij ] be an m × n matrix. Then for any element k ∈ R, we define kA = [kaij ].
- Theorem 2 Let A,B and C be matrices of order m× n, and let k, ℓ ∈ R. Then
1) A + B = B + A commutativity
2) (A + B) + C = A + (B + C) associativity
3) k(ℓA) = (kℓ)
4) (k + ℓ)A = kA + ℓA.
Proof. Let A = [aij ] and B = [bij ]. Then
A + B = [aij ] + [bij ] = [aij + bij ] = [bij + aij ] = [bij ] + [aij ] = B + A
as real numbers commute.
Multiplication of Matrices
Let A = [aij ] be an m × n matrix and B = [bij ] be an n × r matrix. The product AB is a matrix C = [cij ] of order m× r, with
Observe that the product AB is defined if and only if the number of columns of A = the number of rows of B.
Note that in this example, while AB is defined, the product BA is not defined. However, for square matrices A and B of the same order, both the product AB and BA are defined.
- Note that if A is a square matrix of order n then AIn = In Also for any d ∈ R, the matrix dIn commutes with every square matrix of order n. The matrices dIn for any d ∈ R are called scalar matrices.
2. In general, the matrix product is not commutative. For example, consider the following two matrices
- Theorem: For any matrix A, we have (At)t =