Identity Matrix Let A=[ai j] be an n ×n matrix. Prove that A In=In A=A .

Name: Problem 56, Chapter 7: Precalculus: Graphical, Numerical, Algebraic
Uploaded: 2019-07-24T23:37:37-08:00
Duration: 5 min 50 s
Channel: Wendy Wang
Description: Identity Matrix Let A=[ai j] be an n ×n matrix. Prove that A In=In A=A .

Identity Matrix Let A=[ai j] be an n ×n matrix. Prove that A...

Key Concepts

Identity Matrix

The identity matrix is a special square matrix in which all the elements on the main diagonal are one and all other elements are zero. This matrix acts as the multiplicative identity in matrix operations, meaning that when any matrix is multiplied by the identity matrix, it remains unchanged.

Matrix Multiplication

Matrix multiplication is an operation where the elements of the rows of the first matrix are multiplied with the corresponding elements of the columns of the second matrix, and then these products are summed to yield a single entry in the resulting matrix. This operation follows specific algebraic rules which ensure that certain matrices, like the identity matrix, have predictable effects on other matrices.

Algebraic Proof Techniques in Linear Algebra

When asserting properties like AI_n = I_nA = A, one typically uses element-wise verification where each element of the resulting matrix is calculated by summing the products of elements from the respective row and column. This approach clearly demonstrates that only the corresponding elements of the original matrix are retained, thereby providing a rigorous proof of the identity property in matrix multiplication.

Transcript

00:01 For this question, we have a, which is an n -by -n matrix, and we're asked to prove that a times in is equal to in times a, which is equal to a.

00:08 Let's remember what in is.

00:11 I -n is an m -by -n matrix such that it's ij's entry, denoted by i -j, is equal to 0 if i is not equal to j, and it's equal to 1 if i is equal to j.

00:25 To visualize, i .n is a matrix.

00:30 Such that we have ones on the diagonals and zero everywhere else.

00:37 Something like this.

00:41 So how do we show a times i n is equal to in equal to a is equal to a? let's show it by steps.

00:48 So let's first show a times i n is equal to a.

00:52 To make it more convenient, let's call this matrix b.

00:57 So two matrices are equivalent if they're equal to each other, if every entry is equal to each other.

01:05 So if we can show that the ij's entry of b is the ij's entry of a, we are done.

01:10 So let's see what is the ij entry of b, b ij, what is it? using the definition, using the formula for matrix multiplication, we see that since b is equal to a times in, b ij is equal to a, i1, times i1j, plus a, i2, plus i2j plus a i3 oh sorry this is a typo this is supposed to be multiplication right here a i2 times i2 j um plus a i3 times i 3 j plus all the way to a i n times i n j so looking at the entries for i n if i is not equal to j, the i ij term will be zero.

02:11 So for every single one of these terms, besides one, the i ij term will be zero.

02:19 So we're left with the only term that's not zero is when i is equal to j.

02:24 And that is the case if we have i j, j, j.

02:33 Because we're always working with the j column and what is the corresponding a term? well, if we're in the j -th column, we must have a -i -j.

02:49 Now, j is equal to j, so this term becomes 1.

02:55 So essentially we're dealing with a -i -j times 1, which is just a -i -j...

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