Linear Algebra by Dr. Shorouk Ossama

References “Elementary Linear Equation”, tenth Edition, Applications Version, Howard Antion, Chris Rorres.

3.1 Matrix definition: We used rectangular arrays of numbers, called matrices, to abbreviate systems of linear equations. However, Rectangular arrays of numbers occur in other contexts as well. For example, the following rectangular array with three rows and seven columns might describe the number of hours that a student spent studying three subjects during a certain week:

A matrix is a rectangular array of numbers A matrix is a rectangular array of numbers. The numbers in the array are called the entries or element in the matrix. Capital letters are usually used to denote matrices.

The matrix with only one column, B = 𝑏 1 ⋮ 𝑏 𝑛 , is called a column vector or a column matrix, The matrix with only one row , a = [ a1 a2 ……. an] is called a row vector or a row matrix.

The size of a matrix is described in terms of the number of rows (horizontal lines) and columns (vertical lines) it contains. For example, 𝟏 𝟐 𝟑 𝟎 −𝟏 𝟒 the matrix has three rows and two columns, so its size is 3 by 2 (written 3x2).

A general m×n matrix, one with m rows and n columns, can be represented by the notation. Where: aij is the element in the ith row and jth column of A; or A = 𝑎 𝑖𝑗 .

Example: For the matrix, A = 2 −3 7 0 , we have: A11 = 2, A12 = -3, A21 = 7 and A22 = 0.

Some Examples of Matrices Identity matrix: A diagonal matrix with all diagonal elements equal to 1 is called an identity matrix ( I ). AI = A , IA = A

Zero matrix: A matrix whose all entries are zero is called a zero matrix A0 = 0 , 0A = 0

Square Matrix: If the number of rows of a matrix is equal to the number of columns of a matrix , that is, , then is called a square matrix.

Upper Triangular Matrix: A matrix for which for all is called an upper triangular matrix. That is, all the elements below the diagonal entries are zero.

Lower Triangular Matrix: A matrix for which for all is called a lower triangular matrix. That is, all the elements above the diagonal entries are zero.

Diagonal matrix: A square matrix with all non-diagonal elements equal to zero is called a diagonal matrix, that is, only the diagonal entries of the square matrix can be non-zero,

Power of Matrices: If A is a square matrix of order nxn, then we write AA as A2, AA2 as A3 and so on. If A is diagonal, as in: Aⁿ = AA∙ ∙ ∙A

Example: If A = 𝟏 𝟐 𝟏 𝟑 Then A3 = 1 2 1 3 1 2 1 3 1 2 1 3 = 11 30 15 41

Operation on Matrices it is desirable to develop an “arithmetic of matrices” in which matrices can be: added, subtracted, and multiplied in a useful way.

Equality Two matrices are defined to be equal if they have the same size and their corresponding entries are equal. The equality of two matrices A = 𝑎 𝑖𝑗 and B = 𝑏 𝑖𝑗 of the same size can be expressed either by writing 𝐴 𝑖𝑗 = 𝐵 𝑖𝑗

Example: Consider the matrices 𝑪= 𝟐 𝟏 𝟎 𝟑 𝟒 𝒙 Find value of x If A = B, then x = 5, but there is no value of x for which since A and C have different sizes.

Example Solve the equation A=B when Since and must have the same elements, if follows that

Multiplication By a Constant If A is any matrix and k is any scalar, then the product kA is the matrix obtained by multiplying each entry of the matrix A by k. The matrix kA is said to be a scalar multiple of A. If A = [aij], then (kA)ij = k (A)ij = k aij

Example: For The Matrices A = 𝟐 𝟑 𝟒 𝟏 𝟑 𝟏 , 𝑩= 𝟎 𝟐 𝟕 −𝟏 𝟑 −𝟓 and C = 𝟗 −𝟔 𝟑 𝟑 𝟎 𝟏𝟐 We have: 2A = 4 6 8 2 6 2 , (−1)𝐵= 0 −2 −7 1 −3 5 and 1 3 C = 3 −2 1 1 0 4

If : And k= 10, then Equally, we can 'factorize' a matrix. Thus

Zero Matrix Any matrix in which every element is zero is called a zero or null matrix. If A is a zero matrix, we can simply write A= 0 A + 0 = 0 + A = A A – A = 0 0 – A = - A A 0 = 0; 0 A = 0

Matrix Sums and Differences If A and B are matrices of the same size, then the sum A + B is the matrix obtained by adding the entries of B to the corresponding entries of A, and the difference A – B is the matrix obtained by subtracting the entries of B from the corresponding entries of A.

If are both mxn matrices, then and

Matrices of different sizes cannot be added or subtracted. In matrix notation, if A = [aij], then B = [bij] have the same size, then: (A+B)ij = (A)ij + (B)ij = aij + bij and (A-B)ij = (A)ij - (B)ij = aij - bij

Example: Consider the matrices A = 𝟐 𝟏 𝟎 −𝟏 𝟎 𝟐 𝟒 −𝟐 𝟕 𝟑 𝟒 𝟎 , B = −𝟒 𝟑 𝟓 𝟐 𝟐 𝟎 𝟑 𝟐 −𝟒 𝟏 −𝟏 𝟓 , C = 𝟏 𝟏 𝟐 𝟐 Then A+B = −2 4 5 1 2 2 7 0 3 4 3 5 A-B = 6 −2 −5 −3 −2 2 1 −4 11 2 5 −5 The expressions A+C, B+C and B-C are undefined.

Example If Then find A+B, B+A and A+2B. We have (by Rule 2).

Also

A+B = B+A

The difference of two matrices is written as which is interpreted as A - B, A + (-1) B, Example: Find A - B, 2A - 3B if: and

The rules of arithmetic as applied to the elements of matrices lead to the following results for matrices for which addition can be defined:

Problems: Find B – C, 2B – 3C, C + B if: and

Thanks