Printable exercises on det(AB) = det(A)det(B) for advanced students

🔢 Topic Summary

The determinant of a matrix is a special number that can be computed from the elements of a square matrix. It reveals important information about the matrix, such as whether the matrix is invertible. A key property of determinants is that the determinant of the product of two matrices is equal to the product of their determinants. That is, given two $n \times n$ matrices $A$ and $B$, the determinant of their product, denoted as $\det(AB)$, is equal to the product of their individual determinants: $\det(A) \cdot \det(B)$. This property simplifies calculations and provides insights into matrix transformations.

This worksheet will help you practice using the rule $\det(AB) = \det(A)\det(B)$ with a variety of problems designed for advanced students. This property is fundamental in linear algebra and has applications in diverse fields such as physics, engineering, and computer science.

🧠 Part A: Vocabulary

Match each term with its correct definition:

1. Term: Determinant
2. Term: Matrix
3. Term: Invertible Matrix
4. Term: Scalar
5. Term: Product of Matrices

Definitions (Unordered):

1. A rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
2. A matrix that has an inverse; that is, there exists another matrix that, when multiplied with it, results in the identity matrix.
3. The result of multiplying two or more matrices together, obtained by a specific set of operations involving the elements of the matrices.
4. A real number that multiplies a matrix or vector.
5. A scalar value that can be computed from the elements of a square matrix and encodes certain properties of the transformation described by the matrix.

✍️ Part B: Fill in the Blanks

Complete the following paragraph using the words provided:

("determinant", "matrices", "product", "square", "scalars")

The _______ of two _______ $A$ and $B$ (of the same size) is a matrix $AB$ found by multiplying rows of A by columns of B. The _______ of a matrix is a special number that can be computed from the elements of a _______ matrix. If $A$ and $B$ are two matrices and $k$ are _______, then det($AB$) = det($A$)det($B$).

🤔 Part C: Critical Thinking

Consider two $n \times n$ matrices, $A$ and $B$. If $\det(A) = 0$, what can you conclude about the invertibility of the matrix $AB$? Explain your reasoning, referencing the property $\det(AB) = \det(A)\det(B)$. Does your conclusion change if $\det(B) = 0$?