Solved problems: Applying the multiplicative property of determinants

📚 Understanding the Multiplicative Property of Determinants

The multiplicative property of determinants is a powerful tool in linear algebra, allowing us to easily calculate the determinant of a product of matrices. It states that for any two square matrices $A$ and $B$ of the same size, the determinant of their product is equal to the product of their determinants. In mathematical notation, this is expressed as:

$\det(AB) = \det(A) \cdot \det(B)$

📜 Historical Background

The concept of determinants dates back to ancient times, with early forms used in China centuries ago. However, the formal study and development of determinants as a mathematical tool began in the 17th century with mathematicians like Seki Kōwa in Japan and Gottfried Wilhelm Leibniz in Europe. The multiplicative property was later formalized as the field of linear algebra matured, providing a fundamental link between matrix multiplication and determinant calculations.

🔑 Key Principles

• 🧮 Fundamental Rule: The determinant of a product of matrices equals the product of the determinants. $\det(AB) = \det(A) \cdot \det(B)$.
• 📏 Square Matrices: The matrices $A$ and $B$ must be square matrices of the same dimensions for the property to hold.
• 🔁 Order Independence: While matrix multiplication is generally not commutative ($AB \neq BA$), the determinant of $AB$ is equal to the determinant of $BA$ since $\det(AB) = \det(A)\det(B) = \det(B)\det(A) = \det(BA)$.
• 💡 Scalar Multiplication: If $A$ is an $n \times n$ matrix and $c$ is a scalar, then $\det(cA) = c^n \det(A)$.
• ➕ No Additive Property: It's important to note that $\det(A + B) \neq \det(A) + \det(B)$ in general. The multiplicative property applies to products, not sums.

✍️ Solved Problems: Applying the Multiplicative Property

Let's look at some examples to see how this property works in practice:

Example 1:

Given matrices $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, find $\det(AB)$.

Solution:

First, calculate the determinants of $A$ and $B$ individually:

$\det(A) = (1)(4) - (2)(3) = 4 - 6 = -2$

$\det(B) = (5)(8) - (6)(7) = 40 - 42 = -2$

Now, apply the multiplicative property:

$\det(AB) = \det(A) \cdot \det(B) = (-2) \cdot (-2) = 4$

Example 2:

Given matrices $A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 \\ 2 & 2 \end{bmatrix}$, find $\det(AB)$.

Solution:

Calculate the determinants of $A$ and $B$:

$\det(A) = (2)(3) - (1)(0) = 6$

$\det(B) = (1)(2) - (-1)(2) = 2 + 2 = 4$

Apply the multiplicative property:

$\det(AB) = \det(A) \cdot \det(B) = 6 \cdot 4 = 24$

Example 3:

Let $A$ be a $3 \times 3$ matrix with $\det(A) = 5$. Find $\det(A^2)$.

Solution:

$A^2 = A \cdot A$, so $\det(A^2) = \det(A \cdot A) = \det(A) \cdot \det(A) = 5 \cdot 5 = 25$.

Example 4:

If $\det(A) = 2$ and $\det(B) = -3$, find $\det(A^{-1}B)$.

Solution:

$\det(A^{-1}B) = \det(A^{-1}) \cdot \det(B)$. Since $\det(A^{-1}) = \frac{1}{\det(A)}$, we have $\det(A^{-1}B) = \frac{1}{2} \cdot (-3) = -\frac{3}{2}$.

Example 5:

Given matrices $A = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 1 \\ 0 & 2 \end{bmatrix}$, find $\det(2AB)$.

Solution:

$\det(A) = (1)(3) - (0)(2) = 3$

$\det(B) = (-1)(2) - (1)(0) = -2$

Since $A$ and $B$ are $2\times 2$ matrices, $2AB$ means we have to multiply by $2$ twice, once for each row. Thus, $\det(2AB) = 2^2 \det(AB) = 4\det(A)\det(B) = 4(3)(-2) = -24$.

Example 6:

Let $A$ be a $2 \times 2$ matrix such that $\det(A) = 4$. Find $\det(3A^2)$.

Solution:

$\det(3A^2) = \det(3A \cdot A) = 3^2 \det(A) \det(A) = 9 \cdot 4 \cdot 4 = 144$.

Example 7:

If $A$ is a $3 \times 3$ matrix and $\det(A) = -2$, what is $\det(A^3)$?

Solution:

$\det(A^3) = \det(A \cdot A \cdot A) = \det(A) \cdot \det(A) \cdot \det(A) = (-2)^3 = -8$.

🎯 Real-World Applications

• ⚙️ Engineering: Used in structural analysis to determine the stability of systems.
• 💻 Computer Graphics: Determinants help calculate transformations and projections in 3D graphics.
• 📈 Economics: Employed in econometric models to analyze relationships between economic variables.

✅ Conclusion

The multiplicative property of determinants is a fundamental concept in linear algebra. Understanding and applying this property simplifies determinant calculations and is essential for various applications in mathematics, science, and engineering. By grasping the core principles and practicing with examples, you can confidently tackle problems involving the determinants of matrix products.