Let's analyze each statement provided in the image and determine whether they are true or false:

1. If a matrix is invertible, then its inverse is unique.

   • True. If a matrix $A$ is invertible, there exists a unique matrix $A^{-1}$ such that $A \times A^{-1} = I$, where $I$ is the identity matrix. Uniqueness is guaranteed by the properties of matrix multiplication.

2. If $A$ and $B$ are invertible, then $(AB)^{-1} = A^{-1}B^{-1}$.

   • False. The correct formula is $(AB)^{-1} = B^{-1}A^{-1}$. The order of multiplication is reversed when taking the inverse of a product of two matrices.

3. All square matrices are invertible.

   • False. Not all square matrices are invertible. A square matrix is invertible only if its determinant is non-zero. A square matrix with a determinant of zero is called "singular" and does not have an inverse.

4. Every identity matrix is invertible.

   • True. The identity matrix $I$ of any size is always invertible, and its inverse is itself, i.e., $I^{-1} = I$.

Conclusion:

The statements that are true are:

• Statement 1: If a matrix is invertible, then its inverse is unique.
• Statement 4: Every identity matrix is invertible.

Would you like more details or have any questions?

Related Questions:

1. What is the definition of an invertible matrix?
2. How do you find the inverse of a 2x2 matrix?
3. What are some examples of non-invertible matrices?
4. Can a rectangular matrix be invertible?
5. How does the determinant relate to matrix invertibility?

Tip:

Remember, for two matrices $A$ and $B$, $(AB)^{-1} = B^{-1}A^{-1}$. The order is reversed when taking the inverse of a product.