Find each product, if possible.[312532][3−6430]\left[ \begin{matrix} \sqrt{3} & 1 \\ 2\sqrt{5} & 3\sqrt{2} \end{matrix} \right] \left[ \begin{matrix} \sqrt{3} & -\sqrt{6} \\ 4\sqrt{3} & 0 \end{matrix} \right]

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 61

Chapter 6, Problem 61

Find each product, if possible.
$[\begin{matrix} \sqrt{3} & 1 \\ 2 \sqrt{5} & 3 \sqrt{2} \end{matrix}] [\begin{matrix} \sqrt{3} & - \sqrt{6} \\ 4 \sqrt{3} & 0 \end{matrix}]$

Verified step by step guidance

Identify the expressions or polynomials that need to be multiplied. This could be binomials, trinomials, or other polynomial forms.

Apply the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial. For example, if multiplying $(a + b)(c + d)$, multiply $a$ by $c$ and $d$, then multiply $b$ by $c$ and $d$.

Write down all the products obtained from the distribution step, ensuring no terms are missed.

Combine like terms by adding or subtracting coefficients of terms that have the same variable parts and exponents.

Simplify the expression fully to write the final product in standard polynomial form, arranging terms in descending order of degree.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials by applying the distributive property. Each term in the first polynomial is multiplied by every term in the second polynomial, and like terms are combined to simplify the result.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property is essential in algebra for expanding expressions and multiplying polynomials by distributing each term across the sum inside parentheses.

Combining Like Terms

After multiplying polynomials, like terms—terms with the same variable raised to the same power—must be combined by adding or subtracting their coefficients. This simplifies the expression into its standard polynomial form.

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Find each product, if possible. See Examples 5–7. <4x2 Matrix>

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Textbook Question

Use the determinant theorems to evaluate each determinant. See Example 4.

$∣ \begin{matrix} 3 & - 6 & 5 & - 1 \\ 0 & 2 & - 1 & 3 \\ - 6 & 4 & 2 & 0 \\ - 7 & 3 & 1 & 1 \end{matrix} ∣$