12. Use the properties of the determinant function to show that if A and B are any n × n matrices, then: A •B is invertible if and only if A and B are both invertible.
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
7.3 #12
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![### Matrix Invertibility and Determinants
**Exercise 12:**
Use the properties of the determinant function to show that if \( A \) and \( B \) are any \( n \times n \) matrices, then \( A \cdot B \) is invertible **if and only if** \( A \) and \( B \) are both invertible.
**Explanation:**
To tackle this exercise, you will need to delve into the fundamental properties of determinants in linear algebra. Recall that for any square matrices \( A \) and \( B \), the following relationship holds:
\[ \det(A \cdot B) = \det(A) \cdot \det(B) \]
Using this property, you can ascertain the invertibility condition.
- **Invertibility Condition:**
- A matrix \( A \) is invertible if and only if \( \det(A) \neq 0 \).
- Similarly, \( B \) is invertible if and only if \( \det(B) \neq 0 \).
Hence, for the product \( A \cdot B \) to be invertible, both \( \det(A) \neq 0 \) and \( \det(B) \neq 0 \) must hold true.
Putting it all together, \( A \cdot B \) is invertible **if and only if** both \( A \) and \( B \) are invertible.
By following this path, you'll be able to demonstrate the invertibility of the matrix product using determinant properties effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcecab317-fab7-42bc-8c65-e82787a88e59%2Fff233e6d-67f7-4dcd-9c5f-d452afd0b310%2Fbyv9o6a_processed.png&w=3840&q=75)
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