Yes, you can multiply any matrix by an identity matrix, and the result will be the original matrix. This is because the identity matrix acts as the multiplicative identity in matrix multiplication, similar to how the number 1 is the multiplicative identity in scalar multiplication.

Here's why:

• The identity matrix $I_n$ of size $n \times n$ has ones on the diagonal and zeros elsewhere.
• When you multiply any $m \times n$ matrix $A$ by the $n \times n$ identity matrix $I_n$, the result is the original matrix $A$.

Example:

If $A$ is a matrix and $I_n$ is the identity matrix of appropriate size, then: $A \times I_n = A$

For a 2x2 Matrix:

Let $A$ be:

a & b \\ c & d \end{pmatrix}$$ And the identity matrix $$I_2$$ is: $$I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ Multiplying $$A$$ by $$I_2$$: $$A \times I_2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a*1 + b*0 & a*0 + b*1 \\ c*1 + d*0 & c*0 + d*1 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} = A$$ ### Conclusion: Multiplying any matrix $$A$$ by an identity matrix of appropriate size $$I$$ leaves $$A$$ unchanged. This property is fundamental in linear algebra and has various applications in mathematical operations and transformations. Would you like more details or have any questions? Here are 5 questions related to this topic: 1. How does matrix multiplication work for non-square matrices? 2. What are the properties of the identity matrix? 3. Can you provide an example of multiplying a 3x3 matrix by the identity matrix? 4. How does the identity matrix relate to solving systems of linear equations? 5. What is the result of multiplying two identity matrices together? **Tip:** When performing matrix multiplications, always ensure the inner dimensions match (i.e., the number of columns in the first matrix equals the number of rows in the second matrix).