Question

If \(\left|\begin{array}{cc}7 a-5 b & 3 c \\ -1 & 2\end{array}\right|=0\), then which of the following is true? (1) \(14 \mathrm{a}+3 \mathrm{c}=5 \mathrm{~b}\) (2) \(14 a-3 c=5 b\) (3) \(14 \mathrm{a}+3 \mathrm{c}=10 \mathrm{~b}\) (4) \(14 \mathrm{a}+10 \mathrm{~b}=3 \mathrm{c}\)

Short answer

(1) \(7a-5b=3c\) (2) \(14a-10b=3c\) (3) \(14a+3c=10b\) (4) \(21a+3c=15b\) Answer: (3) \(14a+3c=10b\)

Step-by-step solution

Write down the given matrix

We are given a matrix: $$\begin{pmatrix}7a-5b & 3c\\-1 & 2\end{pmatrix}$$

Calculate the determinant

To find the determinant of the given \(2\times2\) matrix, we use the formula \(\text{det}(A)=ad-bc\). So the determinant of the given matrix is: $$\text{det}=\left(7a-5b\right)(2)-(-1)(3c)$$

Set the determinant equal to zero

We are given that the determinant of the matrix equals zero, so we have the equation: $$(7a-5b)(2)+3c=0$$

Simplify the equation

Now let's simplify the equation: $$14a-10b+3c=0$$ Rearrange the equation to see which statement is true: $$14a+3c=10b$$

Compare the simplified equation with the given statements

Comparing the result with the given statements, we find that: $$14a+3c=10b$$ matches with statement (3): $$14a+3c=10b$$ So, the answer is: (3) \(14a+3c=10b\)

Key concepts

Matrix Algebra

Matrix algebra is a branch of mathematics, specifically focusing on the manipulation and study of matrices. Matrices are rectangular arrays of numbers arranged in rows and columns. They are crucial tools in various fields, including engineering, physics, computer science, and economics.
One of the key operations in matrix algebra is finding the determinant, which is a special number that can be calculated from its elements. The determinant helps to understand various properties of a matrix, such as whether the matrix is invertible or singular. In simpler terms, a matrix with a determinant of zero is singular, meaning it does not have an inverse. This is closely related to solving systems of equations, especially in linear algebra.
To calculate determinants of matrices, particularly the simple 2x2 type, specific formulas can be used, like the one utilized in the provided exercise. The process requires utilizing the elements of the matrix in a precise order to calculate the determinant according to the standard formula for a 2x2 matrix, which is essentially the product of the diagonal elements minus the product of the non-diagonal elements.

2x2 Matrices

A 2x2 matrix is one of the simplest forms of matrices with two rows and two columns. The general form of a 2x2 matrix can be represented as:

• \[\begin{pmatrix}a & b \c & d\end{pmatrix}\]

In the context of the exercise given, our matrix is:

• \[\begin{pmatrix}7a-5b & 3c \-1 & 2\end{pmatrix}\]

Understanding Determinants:
The determinant of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is found using the formula \(ad - bc\). This determines the linear dependency of the rows or columns within that matrix. If the determinant is zero, as in this exercise, it indicates these rows or columns are linearly dependent, having no distinct solution when related to linear equations.
These fundamental determinants provide insights into the matrix's properties, such as invertibility, which is crucial for solving linear systems when represented as matrix equations.

Linear Equations

Linear equations are equations of the first order, involving only constants and variables raised to the first power. In matrix algebra, systems of linear equations can be expressed in matrix form, often leading to efficient solutions using methods such as Cramer's rule or row reduction.
In our exercise, the determinants zero condition led to the simplification of linear equations represented by the elements in the matrix. Therefore, establishing a relationship between the variables \(a, b,\) and \(c\) through the matrix formulation. Here’s how it unfolded:
The simplified determinant equation is:

• \[14a - 10b + 3c = 0\]

Rearranging it, we arrived at the equation:

• \[14a + 3c = 10b\]

This resulting equation corresponds to a linear equation, where each variable associates linearly with others. Solving this allows us to understand how each variable affects the others. Such linear equations are fundamental in predicting behavior within modeled systems and are pivotal for analyses that require linear relationships. Predictions and solutions derived from these are often applied to real-world problems, including budgeting, planning, and optimizing resource allocations.