Provide a proof for each of the following. Show that I3 A=A...

$$\text { Provide a proof for each of the following.}$$ Show that $I_{3} A=A$ for 2 $A=\left[\begin{array}{rrr}-2 & 4 & 0 \\ 3 & 5 & 9 \\ 0 & 8 & -6\end{array}\right]$ and $I_{3}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$ (This result, along with that of Example 1 , illustrates that the commutative property holds when one of the matrices is an identity matrix.)

$$\text { Provide a proof for each of the following.}$$
Show that $I_{3} A=A$ for 2
$A=\left[\begin{array}{rrr}-2 & 4 & 0 \\ 3 & 5 & 9 \\ 0 & 8 & -6\end{array}\right]$ and $I_{3}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
(This result, along with that of Example 1 , illustrates that the commutative property holds when one of the matrices is an identity matrix.)

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Key Concepts

Commutative Behavior with the Identity Matrix

Although matrix multiplication is generally not commutative, the identity matrix is a special case that does commute with any matrix. This means that when the identity matrix is one of the factors, the order of multiplication does not affect the result, as I · A equals A and A · I equals A.

Identity Matrix

An identity matrix is a square matrix with ones on the main diagonal and zeros in all other positions. It serves as the multiplicative identity in matrix algebra, meaning that for any conformable matrix, multiplying by the identity matrix (from either side) yields the original matrix.

Matrix Multiplication

Matrix multiplication is defined by taking the dot product of the rows of the first matrix with the columns of the second matrix. This operation requires that the number of columns in the first matrix equals the number of rows in the second matrix, and it forms the foundation of many proofs and concepts in linear algebra.

Properties of the Identity Matrix

The critical property of the identity matrix in linear algebra is that it leaves any matrix unchanged when used in multiplication. This property highlights the role of the identity matrix as the neutral element in matrix multiplication, analogous to the number 1 in scalar arithmetic.

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