Department

Mathematics

Course Code

MATA23H3

Professor

Sophie Chrysostomou

Linear Systems and Matrices

Linear Systems

DEFINITION: An m×nlinear system of equations is asystem of mlinear equations in n

variables: a11x1+a12x2+... +a1nxn=b1

a21x1+a22x2+... +a2nxn=b2

.

.

..

.

..

.

..

.

..

.

..

.

.

am1x1+am2x2+... +amnxn=bm

where x1,· · · xnarethe unknowns and aij are constantreal numbersforalli=1,2,· · · ,m,

and j=1,2,· · · n.

EXAMPLE: 2x1+3x2=9

x1−2x2=1

In order to solvesuchasystem weperformin some sequence the following three operations:

1. Switchtwoequations.

2. Multiply an equation byanonzero constant.

3. Replace an equation bythatequation plus amultiple ofanother equation.

www.notesolution.com

DEFINITION:

1. An m×nmatrix is anordered rectangular array,of realnumbers,with mrows and n

columns.

a11 a12 · · · a1n

a21 a22 · · · a2n

.

.

..

.

..

.

..

.

.

am1am2· · · amn

2. Wemaydenote the matrix bygiving it aname,sayA,and write

A=[aij ], where aij is the entry in the ith rowand jth column of A.

3. An m×1matrix

b1

b2

.

.

.

bm

is calledacolumn vectorinRm.

4. A1×nmatrix c1c2· · · cnis calledarow vectorin Rn.

5. Amatrix is called asquarematrix if the number ofits rowsand the number ofits

columns are equal.

6. Asquare matrix Ais calledadiagonal matrix if

aij =0forall i6=j

7. Asquare matrix Ais calledan identity matrix if

aij =0for all i6=j

1for all i=j

8. Asquare matrix Ais calledan upper triangular matrix if

aij =0forall i>j

9. Asquare matrix Ais calledalower triangular matrix if

aij =0forall i<j

2c

2011 bySophie Chrysostomou

www.notesolution.com

Matrices

Addition Subtractionand Scalar Multiplication

DEFINITION: Let A=[aij ]and B=[bij ]bem×nmatrices and kbeascalarthen:

i) A+Bis deﬁned to bethe matrix withijth entry (A+B)ij =aij +bij

ii) A−Bis the matrix with ijth entry (A−B)ij =aij −bij

iii) kAis the matrix with ijth entry (kA)ij =kaij

Amatrix with zero entries only is called, azeromatrix and is denoted byO.

EXAMPLE: Let A=

2−1

3 2

−2 4

and B=

1 3

−2−5

6−7

.Find A+Band 2A−3B.

3c

2011 bySophie Chrysostomou

www.notesolution.com

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