- WELCOME TO INTRODUCTION TO MATRICES.
THE GOALS OF THIS VIDEO ARE TO DEFINE A MATRIX,
IDENTIFY THE DIMENSIONS OF A MATRIX,
IDENTIFY ELEMENTS IN A MATRIX AND ALSO TO PERFORM
MATRIX ADDITION, SUBTRACTION, AND SCALER MULTIPLICATION.
A MATRIX IS A RECTANGULAR ARRANGEMENT OR ARRAY OF NUMBERS
THAT ARE OFTEN CALLED ELEMENTS.
HERE ARE FOUR EXAMPLES OF MATRICES.
NOTICE THAT USUALLY CAPITAL LETTERS REPRESENT A MATRIX
AND THE ARRANGEMENT OF NUMBERS ARE ENCLOSED IN SQUARE BRACKETS
AS WE SEE HERE.
EACH MATRIX HAS A DIMENSION
AND THE DIMENSION M X N OF A MATRIX IDENTIFIES
HOW MANY ROWS AND COLUMNS A SPECIFIC MATRIX HAS.
THE NUMBER OF ROWS IS M AND THE NUMBER OF COLUMNS IS N.
AND THAT'S IMPORTANT BECAUSE THIS FIRST NUMBER HERE
WILL BE ROWS AND THE SECOND NUMBER HERE WILL BE COLUMNS.
SO IF WE TAKE A LOOK AT MATRIX "A",
ROWS GO FROM LEFT TO RIGHT SO THIS MATRIX HAS 1, 2, 3 ROWS.
SO IF WE COUNT DOWN WE CAN SEE WE WOULD HAVE THREE ROWS.
AND THIS MATRIX HAS TWO COLUMNS 1, 2.
SO THIS IS A 3 x 2 MATRIX, THREE ROWS AND TWO COLUMNS.
IF WE TAKE A LOOK AT MATRIX B
THIS HAS 1, 2 ROWS AND 1, 2, 3 COLUMNS,
THEREFORE THIS IS A 2 x 3 MATRIX.
SO HERE YOU CAN SEE WHY IT'S SO IMPORTANT
THAT THE ROWS COME FIRST AND THE COLUMNS ARE SECOND
BECAUSE A 3 x 2 IS QUITE DIFFERENT THAN A 2 x 3 MATRIX.
MATRIX C HAS ONE ROW AND 1, 2, 3 COLUMNS,
THEREFORE IT'S A 1 x 3 MATRIX SOMETIMES CALLED A ROW MATRIX
BECAUSE IT HAS A SINGLE ROW
AND MATRIX D HAS 1, 2 ROWS AND 1 COLUMN,
THEREFORE IT IS A 2 x 1 MATRIX
AND THIS CAN ALSO BE CALLED A COLUMN MATRIX.
SO ONCE AGAIN, IF WE'RE LOOKING AT ANY MATRIX,
IF WE COUNT DOWN WE'LL BE COUNTING THE NUMBER OF ROWS
AND IF WE COUNT ACROSS WE'LL BE COUNTING THE NUMBER OF COLUMNS
AS WE SEE HERE.
LET ME GO AHEAD AND CLEAR THIS HIGHLIGHTING.
SINCE EACH ELEMENT IN A MATRIX HAS A UNIQUE LOCATION
BASED UPON IT'S ROW AND COLUMN
WE CAN IDENTIFY THE LOCATION OF A SPECIFIC ELEMENT
USING THIS NOTATION HERE
WHERE THE FIRST NUMBER WOULD BE THE ROW
AND THE SECOND NUMBER WOULD BE THE COLUMN.
SO THIS ELEMENT HERE IS IN ROW 3 COLUMN 1.
THIS ELEMENT HERE IS IN ROW 2 COLUMN 3 AND SO ON.
NOW SOMETIMES YOU'LL SEE COMMAS IN BETWEEN THE ROW AND COLUMN
AND SOMETIMES YOU WON'T.
IT JUST DEPENDS ON THE TEXT BOOK.
SO IF WE WANT TO IDENTIFY
THESE FOUR ELEMENTS IN THIS MATRIX LOOKING AT "A" SUB 2 3.
THIS REPRESENTS THE NUMBER IN ROW 2 COLUMN 3 OF MATRIX "A".
SO HERE'S ROW 1, HERE'S ROW 2, COLUMN 1, 2, 3
SO "A" SUB 2 3 IS EQUAL TO 5.
"A" SUB 3 2 WOULD BE THE ELEMENT IN ROW 3 COLUMN 2.
SO HERE'S ROW 1, ROW 2, ROW 3 AND THEN COLUMN 1, COLUMN 2.
SO "A" SUB 3 2 IS EQUAL TO 4.
"A" SUB 1 2 WOULD BE THE ELEMENT IN ROW 1 COLUMN 2.
HERE'S ROW 1, COLUMN 1, COLUMN 2.
SO THIS IS EQUAL TO 2.
AND THEN LASTLY "A" SUB 4
2 WOULD BE THE ELEMENT IN ROW 4 COLUMN 2
AND NOTICE THAT THIS MATRIX DOES NOT HAVE A ROW 4,
THEREFORE THIS ELEMENT IS NOT IN THE MATRIX.
NOW LET'S TALK ABOUT HOW WE CAN ADD AND SUBTRACT MATRICES.
THE FIRST THING WE NEED TO BE AWARE OF IS THAT
THE SUM OF TWO MATRICES CAN ONLY BE FOUND IF BOTH MATRICES
HAVE THE SAME DIMENSION.
IF THEY DON'T HAVE THE SAME DIMENSION
THEY CANNOT BE ADDED OR SUBTRACTED.
SO ONCE WE KNOW THAT TWO MATRICES HAVE THE SAME DIMENSION
TO DETERMINE THE SUM WE JUST ADD THE CORRESPONDING ELEMENTS.
AND SIMILARLY, WHEN WE WANT TO FIND
THE DIFFERENCE OF TWO MATRICES
AS LONG AS THEY HAVE THE SAME DIMENSION TO FIND THE DIFFERENCE
WE'LL JUST SUBTRACT THE CORRESPONDING ELEMENTS.
SO LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE OF EXAMPLES.
FOR NUMBER ONE, WE WANT TO SUM MATRIX "A" AND MATRIX B
BUT MATRIX "A" IS A 2 x 2 MATRIX, 2 ROWS AND 2 COLUMNS,
AND MATRIX B IS A 2 x 3 MATRIX.
THESE DON'T HAVE THE SAME DIMENSION
THEREFORE WE CANNOT DETERMINE THIS SUM.
FOR NUMBER 2, WE WANT THE SUM OF MATRIX "A" AND MATRIX C.
WELL MATRIX C IS ALSO A 2 x 2 SO WE CAN FIND THE SUM.
LET'S GO AHEAD AND WRITE THIS OUT.
SO THE RESULT WILL HAVE THE SAME DIMENSION
AND NOW WE JUST ADD THE CORRESPONDING ELEMENTS
OR THE ELEMENTS IN THE SAME POSITIONS.
SO WE'LL HAVE 4 + -3 THAT WOULD BE 1. 7 + -7 THAT'S 0,
AND 2 + 5 THAT WOULD BE 7, AND -1 + 6 = 5.
SO AS YOU CAN SEE IT'S PRETTY STRAIGHT FORWARD.
FOR NUMBER THREE, WE WANT MATRIX C MINUS MATRIX "A".
AGAIN, THEY HAVE THE SAME DIMENSIONS
SO LET'S GO AHEAD AND DO THAT.
WE'LL HAVE MATRIX C - MATRIX "A".
OF COURSE, WHEN WE'RE SUBTRACTING
WE DO HAVE TO BE CAREFUL OF OUR SIGNS.
SO FOR ROW 1 COLUMN 1 WE'LL HAVE -3 - 4 THAT'S GOING TO BE -7.
IN ROW 1 COLUMN 2 WE'LL HAVE -7 - 7 THAT'S -14.
FOR ROW 2 COLUMN 1 WE HAVE 5 - 2 WHICH IS 3,
AND ROW 2 COLUMN 2 WE HAVE 6 - - 1
WHICH BECOMES 6 + 1
SO WE HAVE 7.
LET'S TALK ABOUT SCALAR MULTIPLICATION NOW.
THIS IS GOING TO REMIND YOU OF DISTRIBUTION
WHICH YOU LEARNED IN YOUR FIRST ALGEBRA CLASS.
THE PRODUCT OF A SCALER OR A REAL NUMBER IN A MATRIX
IS EQUAL TO THE SCALER TIMES EACH ELEMENT IN THE MATRIX.
SO IF THIS IS MATRIX B AND WE WANT 5 x B
THIS REPRESENTS SCALER MULTIPLICATION.
SO WE WOULD HAVE 5 x MATRIX B.
SO WE'RE GOING TO MULTIPLY EACH ELEMENT OR EACH NUMBER
IN THIS MATRIX BY 5.
THE DIMENSION WILL STAY THE SAME.
SO FOR ROW 1 COLUMN 1 WE'LL HAVE 5 x -1 THAT'S -5.
FOR ROW 1 COLUMN 2 WE'LL HAVE 5 x 2 THAT'S 10,
AND THEN ROW 1 COLUMN 3, 5 x 8 WOULD BE 40 AND SO ON.
5 x -3, 5 x -7, AND 5 x 1.
AND THAT'S IT FOR SCALER MULTIPLICATION.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE EXAMPLE.
SOMETIMES. YOU WILL BE ASKED TO DETERMINE A SUM
OR DIFFERENCE OF TWO MATRICES
COMBINED WITH SCALER MULTIPLICATION.
SO ON NUMBER ONE WE WANT 4 x MATRIX B + 2 x MATRIX D.
LETS FIRST CHECK THE DIMENSIONS OF THESE TWO MATRICES.
MATRIX B WOULD BE 2 ROWS BY 1 COLUMN
AND MATRIX D IS ALSO 2 ROWS BY 1 COLUMN.
SO WE CAN GO AHEAD AND FIND THIS SUM.
LET'S GO AHEAD AND WRITE IT OUT.
WE'RE GOING TO HAVE 4 x MATRIX B + 2 x MATRIX D.
LET'S GO AHEAD AND APPLY THE SCALER MULTIPLICATION FIRST.
SO WE'LL HAVE 16 AND 28 + 0 AND -6.
SO THE FINAL SUM WILL BE 16 + 0 THAT'S 16.
28 + -6 WILL GIVE US 22.
AND NUMBER TWO, WE HAVE MATRIX C - 3 x MATRIX "A".
BOTH MATRIX C AND "A" ARE 2 x 2 MATRICES
SO WE CAN GO AHEAD AND FIND THIS DIFFERENCE.
LET'S GO AHEAD AND WRITE MATRIX C - 3 x MATRIX "A"
WHICH IS [4 7 2 -1].
FIRST, WE'LL PERFORM THE SCALER MULTIPLICATION
AND THEN WE'LL SUBTRACT.
SO WE'RE GOING TO HAVE [12 21 6 -3].
NOW NOTICE I MULTIPLIED BY A 3
AND KEPT IT AS A DIFFERENCE.
NOW WE'LL GO AHEAD AND SUBTRACT CORRESPONDING ELEMENTS.
-3 - 12 IS -15, -7 - 21 THAT'S - 28, 5 - 6 IS -1
AND 6 - -3 BECOMES 6 + 3 OR 9.
OKAY. THAT WILL DO IT FOR THIS VIDEO.
I HOPE YOU FOUND IT HELPFUL.
THANK YOU FOR WATCHING.