What is the product of a matrix and its inverse? a. Zero matrix b. Identity matrix c. Transpose matrix d. Diagonal matrix

Understanding the Problem

We need to determine what the product of a matrix and its inverse is from the given options.

Relevant Concepts

In linear algebra, the product of a square matrix $A$ and its inverse $A^{-1}$ is defined by the following property:

$A \cdot A^{-1} = I$

where $I$ is the identity matrix.

Analysis and Detail

1. If $A$ is an $n \times n$ matrix and it is invertible, then:

   • The inverse of $A$, denoted $A^{-1}$, is also an $n \times n$ matrix.
   • The product of $A$ and $A^{-1}$ results in the identity matrix $I$.

2. The identity matrix has ones on the diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix multiplication, similar to how the number 1 functions in regular multiplication.

3. The other options:

   • Zero matrix: Any matrix multiplied by the zero matrix results in the zero matrix, not the identity.
   • Transpose matrix: The transpose of a matrix is a different entity and does not relate to the product with its inverse.
   • Diagonal matrix: While a diagonal matrix can have an inverse, its product with the inverse of a different matrix wouldn't yield an identity matrix unless specified.

Verify and Summarize

Since $A \cdot A^{-1} = I$, we confirm that the product yields the identity matrix.

Final Answer

The product of a matrix and its inverse is b. Identity matrix.