Name: \{ert period:J

YAyr,r,r*a*rwMatrix Addition Subtraction. & Scalar Multiplication

Let A and B be rnatrices of the same dimension m x n and let c be any real number.

t\a\viceS I$\Uttr Iheve r$.eJsa$\€ II\$\etsisrsl

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Addition A + b = to,, + b,3 $$-"' "[ffi:*"tSubtraction A-B =t6ii b,3 {t $.rb\xac\ cor*utponar.,qen\\\es \{\ \\a\x.x R 'rtxom sna\xrx F\ "

ScalarMultiplication cA = [co.,,rt

* \\u\ip\q eae\r etftxq .r-1\\a\\\x" N bg \\€Corrr\anh"

Examples: civen the matrices below perform the indicated operations.

^=E I l-1r = l-slz[t il.Bl=titifl=Li il2::

t.i il r' il=E] TI]=B il[r- zJ \-r ^rd E'- | 'L-k)3. 4A-38

;'rHqUIp--;Hp,F,-L n

,1. A+ B

lr.+ b

Matrix Multiolication

Matricbi can only be multiplied teryether if:

o A matrix with m x n dimensions is being multiplied by a matrix with nx kdimensions.

r The resulting product will be a matrix with m x k dimensions.Example: Civen A & B U"t"*ffiXmatrix AB.

e=[_1r ]rfuWfi. 5214rl 4x

Entry lnner Product of; Value Product Matrix

Cu gq8 i i1 '. .+ ---\

Crr 9ffr9;r t(b)+t'i=;=\l r' \.1

Crt t9''elt? ?q \(t)+ ,,,=13\l

Ctr 15 ilr?; i1 + =\

C.r 16 fi{t?g;r + =*ia r\\1 1_h-5

C.r b $Ir? ;q + :-^) -\ \1 1-5

t -5 -1-

[l \ a\

lnverses of Matrices

en \dQn\ is an m x a'matrix in which each main diagonal entry is a 1and for which I other entries are 0.

Examples:

t. Muttipty, = [ l] o, , = [j, i f]. wr.r"t do you notice? why would matrixJr Nn rden\,\ A be callea an;iA6ntity" matrixl

a " , * b "*]l$tdoes no\

W** g : iix r'f -l$:;5 fi:[l = B : I

Example:

:) N ? S aYe \\rveYSeS oQ eac\r o\hev'

L\\\il-r [-r oLI ll Lt :,

1. Verify that a = [! ]] ,ru , = [-1 ;1N r=E []tl il =83'l[:] l:\::il =t* 3rJ=L'Ie x=L3 )ltlI=E$lllf itli:'l8l =f;l'"TJ=L:ll

To verify that one matrix is the inverse of another:

=Fi'.N=a*=tT"Y

How to Find the lnverse of a Matrix

The most efficient way to find the identity of an rn x n matrix is to use row operations asdemonstrated in the example below:

tL -2 -41e=lz -3 -61[-s 6 1sl

-,[l I I\]. 1!l--7 L; o ) lac\Jq-l. Find l-1

n -L -\ t\ o\, -1 -G \o \tl ; \5 loo

+ [l J i\iiil:;:*[oo I+ I-l : t \"3 i "Ya.-lR,