- WELCOME TO A VIDEO
ON THE IDENTITY MATRIX,
AND THE GOAL OF THIS VIDEO IS
TO DEFINE AN IDENTITY MATRIX.
AN END BY END SQUARE MATRIX
WITH A MAIN DIAGONAL OF ONES
AND ALL OTHER ELEMENTS ZEROS
IS CALLED AN IDENTITY MATRIX
DENOTED WITH A CAPITAL LETTER
"I".
AND SINCE THE IDENTITY MATRIX
IS A SQUARE MATRIX,
WE USUALLY DENOTE IT
WITH A SUBSCRIPT HERE
FOR A 2 X 2, 3 X 3, 4 X 4
IDENTITY MATRIX, AND SO-ON.
SO, AGAIN, THE MAIN DIAGONAL
WILL ALWAYS BE ONES
AND ALL OTHER ELEMENTS ZEROS.
SO IF WE HAVE
AN IDENTITY MATRIX
THAT HAS SOME SPECIAL
PROPERTIES--
AND YOU CAN ALMOST THINK OF IT
AS A ONE IN ALGEBRA.
WHENEVER YOU MULTIPLY
SOMETHING BY ONE
IT DOESN'T CHANGE.
SO FOR EXAMPLE,
IF WE HAVE A MATRIX A
THAT IS AN END BY END MATRIX
THEN THE IDENTITY MATRIX x A
WILL = A,
AND A x THE IDENTITY MATRIX
= A AS WELL
IF THAT MULTIPLICATION
IS POSSIBLE.
AND IF A IS AN END BY END
SQUARE MATRIX
THEN A x I WILL = I x A,
WHICH WILL ALWAYS = A.
LET'S GO AHEAD AND PERFORM
SOME MULTIPLICATION
WITH IDENTITY MATRIX
TO GET A BETTER FEEL FOR THIS.
SO FOR EXAMPLE, HERE WE HAVE
A 2 X 2 ANIMATED MATRIX
TIMES ANOTHER 2 X 2 MATRIX.
REMEMBER THIS IS A 2 X 2,
THIS IS A 2 X 2.
SINCE THESE TWO NUMBERS
ARE EQUAL
THE MULTIPLICATION
IS POSSIBLE,
AND RESULT
IS ANOTHER 2 X 2 MATRIX.
SO TO FIND THE ELEMENT
IN ROW ONE, COLUMN ONE,
WE MULTIPLY ROW ONE
BY COLUMN ONE.
SO 1 x 4 WOULD BE 4 + 0 x -5
THAT'S 4.
TO FIND THE ELEMENT
IN ROW ONE, COLUMN TWO,
WE MULTIPLY ROW ONE
IN THE FIRST MATRIX
TIMES COLUMN TWO
IN THE SECOND.
WE'D HAVE 1 x 7 + 0 x -2
WHICH = 7.
AND THEN HERE IN ROW TWO,
COLUMN ONE,
WE MULTIPLY ROW TWO
TIMES COLUMN ONE,
0 x 4 IS 0 AND 1 x -5.
AND THEN FOR ROW TWO,
COLUMN TWO
WE HAVE 0 x 7 WHICH IS 0
AND 1 x -2 WHICH IS -2.
AND SO THE THING TO NOTICE
HERE
IS HERE'S THE IDENTITY MATRIX
TIMES THE MATRIX A,
AND THE RESULT IS MATRIX A
AGAIN.
AND NOTICE
IN THE SECOND PRODUCT
WE JUST CHANGED
THE ORDER OF MULTIPLICATION.
AND LET'S GO AHEAD AND SHOW
THE RESULT WILL BE THE SAME.
FOR THIS ELEMENT HERE
WE'D HAVE 4 x 1 THAT'S 4 + 0.
FOR ROW ONE, COLUMN TWO
WE'D HAVE 4 x 0 WHICH IS 0
AND 7 x 1.
AND ROW TWO, COLUMN ONE
WE'D HAVE -5 x 1
THAT'S -5 + -2 x 0, -5.
AND THEN LASTLY, WE'D HAVE
ROW TWO TIMES COLUMN TWO,
-5 x 0 + -2 x 1.
AND, AGAIN, WE CAN SEE
THE RESULT IS THE SAME.
THIS IS MATRIX A
x THE IDENTITY,
THE RESULT, AGAIN,
IS MATRIX A.
SO MULTIPLICATION
IS COMMUTATIVE
IF WE MULTIPLY A SQUARE MATRIX
TIMES IDENTITY MATRIX.
NOW LET'S GO AHEAD
AND TRY ONE MORE HERE.
AGAIN, THIS IS THE IDENTITY
MATRIX.
IF WE CALL THIS MATRIX B
WE WOULD EXPECT THE RESULT
TO BE MATRIX B.
LET'S GO AHEAD AND CHECK
THIS ONE ON THE CALCULATOR.
LET'S GO AHEAD
AND PRESS SECOND MATRIX.
LET'S GO OVER TO EDIT,
PRESS ENTER.
LET'S MATRIX A THE IDENTITY
MATRIX, SO IT'S A 3 X 3.
WE HAVE 1, 0, 0, 0, 1, 0, 0,
0, AND 1.
THERE'S THE IDENTITY MATRIX.
AND NOW LET'S GO AHEAD
AND ENTER IN MATRIX B,
SECOND MATRIX,
GO OVER TO EDIT, SCROLL TO B,
PRESS ENTER.
THIS IS A 3 X 3.
GO AHEAD AND ENTER
IN THESE ELEMENTS HERE.
AND THERE IT IS.
GO BACK TO THE MAIN SCREEN,
SECOND QUIT.
AND WE'LL MULTIPLY MATRIX B,
SECOND MATRIX,
SCROLL TO B, ENTER,
TIMES THE IDENTITY MATRIX
IS IN MATRIX A.
SO SECOND MATRIX,
ENTER, ENTER,
AND WE CAN SEE THE RESULT
IS MATRIX B AS WE EXPECTED.
SO NOW YOU CAN PROBABLY TELL
WHY YOU CAN THINK
OF THE IDENTITY MATRIX
AS THE NUMBER 1 IN ALGEBRA.
BECAUSE WHEN YOU MULTIPLY IT
THE RESULT IS THE SAME.
OKAY. IN THE NEXT VIDEO WE'LL
TALK ABOUT INVERSE MATRICES.
THANK YOU FOR WATCHING.
