Chapter 4: Q4. (page 185)

Find a counterexample to disprove the following statement.
Two different matrices can never have the same determinant.

Short Answer

Expert verified

The matrix $A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ and $B [\begin{matrix} 8 & 3 \\ 5 & 2 \end{matrix}]$ disprove the statement “Two different matrices can never have the same determinant.”.

Step by step solution

- Assume the matrices

In order to disprove the statement “Two different matrices can never have the same determinant” consider two $2 \times 2$ matrices $A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ and $B [\begin{matrix} 8 & 3 \\ 5 & 2 \end{matrix}]$ and calculate there respective determinants.

-Define a determinant of second order matrix

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is., $|\begin{matrix} a & b \\ c & d \end{matrix}| = a d - b c$ .

- Find the determinants

Calculate the determinants of matrix A and B and observe if they provide the same value.

$\begin{matrix} | A | = |\begin{matrix} \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \end{matrix}| \\ = 1 \times (1) - 0 (0) \\ = 1 \end{matrix}$

And,

$\begin{matrix} | B | = |\begin{matrix} \begin{matrix} 8 & 3 \\ 5 & 2 \end{matrix} \end{matrix}| \\ = 8 \times (2) - 3 (5) \\ = 16 - 15 \\ = 1 \end{matrix}$

Clearly, A and B are two different matrices since their corresponding values are different and also they have same determinants which disprove the statement that “Two different matrices can never have the same determinant”.

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Find a counterexample to disprove the following statement.
Two different matrices can never have the same determinant.

Short Answer

Step by step solution

- Assume the matrices

-Define a determinant of second order matrix

- Find the determinants

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Find a counterexample to disprove the following statement.Two different matrices can never have the same determinant.

Short Answer

Step by step solution

- Assume the matrices

-Define a determinant of second order matrix

- Find the determinants

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Mechanics Maths

Logic and Functions

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Find a counterexample to disprove the following statement.
Two different matrices can never have the same determinant.