3 X 3 DETERMINANTS DIAGONALS METHOD 5 2 7 -1 4 0 3 0 6.

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Presentation transcript:

3 X 3 DETERMINANTS DIAGONALS METHOD 5 2 7 -1 4 0 3 0 6

DIAGONALS METHOD COPY 1ST TWO COLUMNS 5 2 7 -1 4 0 3 0 6

DIAGONALS METHOD COPY 1ST TWO COLUMNS 5 2 7 -1 4 0 3 0 6 5 -1 3

DIAGONALS METHOD COPY 1ST TWO COLUMNS 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0

DIAGONALS METHOD MULTIPY DOWN !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0

DIAGONALS METHOD MULTIPY DOWN !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0

DIAGONALS METHOD MULTIPY DOWN !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120

DIAGONALS METHOD MULTIPY DOWN !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120

DIAGONALS METHOD MULTIPY DOWN!! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120 +0

DIAGONALS METHOD MULTIPY DOWN !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120 +0

DIAGONALS METHOD MULTIPY DOWN !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120 +0 0+

DIAGONALS METHOD MULTIPY DOWN!! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120 +0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120 +0 0+ =

DIAGONALS METHOD MULTIPY DOWN !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 120 +0 0+ = 120

DIAGONALS METHOD 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 5 2 84 -1 4 0 -1 4 3 0 6 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 5 2 84 + 0 -1 4 0 -1 4 3 0 6 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 5 2 84 + 0 -1 4 0 -1 4 3 0 6 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 + (-12) DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 + (-12) = DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 + (-12) = 72 DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 + (-12) = 72 DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 5 2 84 + 0 + (-12) -1 4 0 -1 4 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 + (-12) = 72 ALWAYS: DOWN-UP DOWN = 120

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 + (-12) = 72 ALWAYS: DOWN-UP DOWN = 120 120 -

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 + (-12) = 72 ALWAYS: DOWN-UP DOWN = 120 120 - 72

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 + (-12) = 72 ALWAYS: DOWN-UP DOWN = 120 120 - 72 =

DIAGONALS METHOD MULTIPLY UP !! 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 5 2 7 -1 4 0 3 0 6 5 2 -1 4 3 0 84 + 0 + (-12) = 72 ALWAYS: DOWN-UP DOWN = 120 120 - 72 = 48

3 X 3 DETERMINANTS EXPANSION BY MINORS METHOD 5 2 7 -1 4 0 3 0 6

3 X 3 DETERMINANTS FINDING A MINOR 5 2 7 -1 4 0 3 0 6 5 2 7 -1 4 0 3 0 6 EXAMPLE 1: THE MINOR OF ELEMENT ‘5’: ROW 1,COLUMN 1

FINDING A MINOR 5 2 7 CROSS OUT ROW AND COLUMN CONTAINING ‘5’ -1 4 0 EXAMPLE 1: THE MINOR OF ELEMENT ‘5’:ROW 1,COLUMN 1 5 2 7 -1 4 0 3 0 6 CROSS OUT ROW AND COLUMN CONTAINING ‘5’

FINDING A MINOR CROSS OUT ROW AND COLUMN CONTAINING ‘5’ 5 2 7 -1 4 0 EXAMPLE 1: THE MINOR OF ELEMENT ‘5’:ROW 1,COLUMN 1 5 2 7 -1 4 0 3 0 6 CROSS OUT ROW AND COLUMN CONTAINING ‘5’

FINDING A MINOR 5 2 7 CROSS OUT ROW AND COLUMN CONTAINING ‘5’ -1 4 0 EXAMPLE 1: THE MINOR OF ELEMENT ‘5’:ROW 1,COLUMN 1 5 2 7 -1 4 0 3 0 6 CROSS OUT ROW AND COLUMN CONTAINING ‘5’

FINDING A MINOR EXAMPLE 1: THE MINOR OF ELEMENT ‘5’:ROW 1,COLUMN 1 5 2 7 -1 4 0 3 0 6 THE REMAINING ELEMENTS ARE CALLED THE MINOR MATRIX

FINDING A MINOR THE REMAINING ELEMENTS ARE CALLED THE MINOR 5 2 7 EXAMPLE 1: THE MINOR OF ELEMENT ‘5’:ROW 1,COLUMN 1 THE REMAINING ELEMENTS ARE CALLED THE MINOR MATRIX => 5 2 7 -1 4 0 3 0 6 4 0 0 6

FINDING A MINOR EXAMPLE 2: THE MINOR OF ELEMENT ‘2’:ROW 1,COLUMN 2 5 2 7 -1 4 0 3 0 6

FINDING A MINOR EXAMPLE 2: THE MINOR OF ELEMENT ‘2’:ROW 1,COLUMN 2 5 2 7 -1 4 0 3 0 6 CROSS OUT ROW AND COLUMN CONTAINING ‘2’

FINDING A MINOR EXAMPLE 2: THE MINOR OF ELEMENT ‘2’:ROW 1,COLUMN 2 5 2 7 -1 4 0 3 0 6 CROSS OUT ROW AND COLUMN CONTAINING ‘2’

FINDING A MINOR EXAMPLE 2: THE MINOR OF ELEMENT ‘2’:ROW 1,COLUMN 2 CROSS OUT ROW AND COLUMN CONTAINING ‘2’ 5 2 7 -1 4 0 3 0 6

FINDING A MINOR EXAMPLE 2: THE MINOR OF ELEMENT ‘2’:ROW 1,COLUMN 2 THE REMAINING ELEMENTS ARE CALLED THE MINOR MATRIX => 5 2 7 -1 4 0 3 0 6 -1 0 3 6

FINDING A MINOR EXAMPLE 3: THE MINOR OF ELEMENT ‘7’:ROW 1,COLUMN 3 CROSS OUT ROW AND COLUMN CONTAINING ‘7’ 5 2 7 -1 4 0 3 0 6

FINDING A MINOR EXAMPLE 3: THE MINOR OF ELEMENT ‘7’:ROW 1,COLUMN 3 CROSS OUT ROW AND COLUMN CONTAINING ‘7’ 5 2 7 -1 4 0 3 0 6

FINDING A MINOR EXAMPLE 3: THE MINOR OF ELEMENT ‘7’:ROW 1,COLUMN 3 CROSS OUT ROW AND COLUMN CONTAINING ‘7’ 5 2 7 -1 4 0 3 0 6

FINDING A MINOR EXAMPLE 3: THE MINOR OF ELEMENT ‘7’:ROW 1,COLUMN 3 THE REMAINING ELEMENTS ARE CALLED THE MINOR MATRIX => 5 2 7 -1 4 0 3 0 6 -1 4 3 0

EXPANSION BY MINORS EACH ELEMENT HAS A SIGN ASSOCIATED WITH IT FOR THE EXPANSION BY MINORS METHOD + - + - + - 5 2 7 -1 4 0 3 0 6

EXPANSION BY MINORS 5 2 7 + - + -1 4 0 - + - 3 0 6 PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 + - + - + -

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 4 0 0 6 =5 + - + - + - + - +

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH LEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS 4 0 0 6 =5 -1 0 3 6 - 2 5 2 7 -1 4 0 3 0 6 + - + - + - + - +

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 -1 4 3 0 4 0 0 6 -1 0 3 6 = 5 - 2 +7 + - + - + - + - +

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 -1 4 3 0 4 0 0 6 -1 0 3 6 = 5 - 2 +7 + - + - + - + - + = 5(24-0)

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS -1 4 3 0 5 2 7 -1 4 0 3 0 6 4 0 0 6 -1 0 3 6 = 5 - 2 +7 + - + - + - + - + = 5(24 - 0) – 2(-6 - 0)

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 -1 4 3 0 4 0 0 6 -1 0 3 6 = 5 - 2 +7 + - + - + - + - + = 5(24 - 0) – 2(-6 - 0) + 7(0 – 12)

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 -1 4 3 0 4 0 0 6 -1 0 3 6 = 5 - 2 +7 = 5(24 - 0) – 2(-6 - 0) + 7(0 – 12) + - + - + - + - + = 5(24) – 2(-6) + 7(-12)

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 -1 4 3 0 4 0 0 6 -1 0 3 6 = 5 - 2 +7 + - + - + - + - + = 5(24 - 0) – 2(-6 - 0) + 7(0 – 12) = 5(24) – 2(-6) + 7(-12) = 120 + 12 - 84

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 -1 4 3 0 4 0 0 6 -1 0 3 6 = 5 - 2 +7 = 5(24 - 0) – 2(-6 - 0) + 7(0 – 12) + - + - + - + - + = 5(24) – 2(-6) + 7(-12) = 120 + 12 - 84 = 48

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT’S MINOR MATRIX AND ADD THE RESULTS EXAMPLE #2 5 2 7 -1 4 0 3 0 6 + - + - + -

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT'S MINOR MATRIX AND ADDTHE RESULTS EXAMPLE #2 5 2 7 -1 4 0 3 0 6 -1 0 3 6 - 2 + - + - + - + - +

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT'S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 -1 0 3 6 +4 -2 5 7 3 6 + - + - + - + - +

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT'S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 5 7 -1 0 -1 0 3 6 5 7 3 6 = -2 + 4 -0 + - + - + - + - +

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT'S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 5 7 -1 0 -1 0 3 6 5 7 3 6 = -2 + 4 -0 + - + - + - + - + = -2(-6 - 0) +4(30 - 21) - 0(0 – (-7))

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT'S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 5 7 -1 0 -1 0 3 6 5 7 3 6 = -2 + 4 -0 + - + - + - + - + = -2(-6 - 0) +4(30 - 21) - 0(0 – (-7)) = -2(-6) + 4(9) - 0(-7)

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT'S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 5 7 -1 0 -1 0 3 6 5 7 3 6 = -2 + 4 -0 + - + - + - + - + = -2(-6 - 0) +4(30 - 21) - 0(0 – (-7)) = -2(-6) + 4(9) - 0(-7) = 12 + 36- 0

EXPANSION BY MINORS PICK ANY ROW OR COLUMN AND MULTIPLY EACH ELEMENT WITH IT’S RELATED SIGN BY THAT ELEMENT'S MINOR MATRIX AND ADD THE RESULTS 5 2 7 -1 4 0 3 0 6 5 7 -1 0 -1 0 3 6 5 7 3 6 = -2 + 4 -0 + - + - + - + - + = -2(-6 - 0) +4(30 - 21) - 0(0 – (-7)) = -2(-6) + 4(9) - 0(-7) = 12 + 36- 0 = 48

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