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This module introduces the identity matrix and its properties.

When multiplying numbers, the number 1 has a special property: when you multiply 1 by any number, you get that same number back. We can express this property as an algebraic generalization:

1 x = x

The matrix that has this property is referred to as the identity matrix .

Definition of identity matrix

The identity matrix , designated as [ I ] , is defined by the property: [ A ] [ I ] = [ I ] [ A ] = [ A ]

Note that the definition of [I] stipulates that the multiplication must commute —that is, it must yield the same answer no matter which order you multiply in. This is important because, for most matrices, multiplication does not commute.

What matrix has this property? Your first guess might be a matrix full of 1s, but that doesn’t work:

1 2 3 4 size 12{ left [ matrix { 1 {} # 2 {} ##3 {} # 4{} } right ]} {} 1 1 1 1 size 12{ left [ matrix { 1 {} # 1 {} ##1 {} # 1{} } right ]} {} = 3 3 7 7 size 12{ left [ matrix { 3 {} # 3 {} ##7 {} # 7{} } right ]} {} so 1 1 1 1 size 12{ left [ matrix { 1 {} # 1 {} ##1 {} # 1{} } right ]} {} is not an identity matrix

The matrix that does work is a diagonal stretch of 1s, with all other elements being 0.

1 2 3 4 size 12{ left [ matrix { 1 {} # 2 {} ##3 {} # 4{} } right ]} {} 1 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} ##0 {} # 1{} } right ]} {} = 1 2 3 4 size 12{ left [ matrix { 1 {} # 2 {} ##3 {} # 4{} } right ]} {} so 1 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} ##0 {} # 1{} } right ]} {} is the identity for 2x2 matrices
2 5 9 π 2 8 3 1 / 2 8 . 3 size 12{ left [ matrix { 2 {} # 5 {} # 9 {} ##π {} # - 2 {} # 8 {} ## - 3 {} # 1/2 {} # 8 "." 3{}} right ]} {} 1 0 0 0 1 0 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} # 0 {} ##0 {} # 1 {} # 0 {} ## 0 {} # 0 {} # 1{}} right ]} {} = 2 5 9 π 2 8 3 1 / 2 8 . 3 size 12{ left [ matrix { 2 {} # 5 {} # 9 {} ##π {} # - 2 {} # 8 {} ## - 3 {} # 1/2 {} # 8 "." 3{}} right ]} {} 1 0 0 0 1 0 0 0 1 size 12{ left [ matrix { 1 {} # 0 {} # 0 {} ##0 {} # 1 {} # 0 {} ## 0 {} # 0 {} # 1{}} right ]} {} is the identity for 3x3 matrices

You should confirm those multiplications for yourself, and also confirm that they work in reverse order (as the definition requires).

Hence, we are led from the definition to:

The identity matrix

For any square matrix, its identity matrix is a diagonal stretch of 1s going from the upper-left-hand corner to the lower-right, with all other elements being 0. Non-square matrices do not have an identity. That is, for a non-square matrix [ A ] , there is no matrix such that [ A ] [ I ] = [ I ] [ A ] = [ A ] .

Why no identity for a non-square matrix? Because of the requirement of commutativity. For a non-square matrix [ A ] you might be able to find a matrix [ I ] such that [ A ] [ I ] = [ A ] ; however, if you reverse the order, you will be left with an illegal multiplication.

Questions & Answers

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4
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x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
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x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
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Need help solving this problem (2/7)^-2
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x+2y-z=7
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-1
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the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
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An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
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lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
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A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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