Solve ([0,3,4],[2,9,6],[3,4,8]) | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\left[\begin{matrix} 0 & 3 & 4 \\2 & 9 & 6 \\3 & 4 & 8 \end{matrix}\right]$$

Calculate Determinant

$-70$

Steps Using Sarrus' Rule

Steps Using Expansion by Minors

Steps Using Sarrus' Rule

Find the determinant of the matrix using the method of diagonals.

$$det(\left(\begin{matrix}0&3&4\\2&9&6\\3&4&8\end{matrix}\right))$$

Extend the original matrix by repeating the first two columns as the fourth and fifth columns.

$$\left(\begin{matrix}0&3&4&0&3\\2&9&6&2&9\\3&4&8&3&4\end{matrix}\right)$$

Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.

$$3\times 6\times 3+4\times 2\times 4=86$$

Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.

$$3\times 9\times 4+8\times 2\times 3=156$$

Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.

$$86-156$$

Subtract $156$ from $86$.

$$-70$$

Steps Using Expansion by Minors

Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).

$$det(\left(\begin{matrix}0&3&4\\2&9&6\\3&4&8\end{matrix}\right))$$

To expand by minors, multiply each element of the first row by its minor, which is the determinant of the $2\times 2$ matrix created by deleting the row and column containing that element, then multiply by the element's position sign.

$$-3det(\left(\begin{matrix}2&6\\3&8\end{matrix}\right))+4det(\left(\begin{matrix}2&9\\3&4\end{matrix}\right))$$

For the $2\times 2$ matrix $\left(\begin{matrix}a&b\\c&d\end{matrix}\right)$, the determinant is $ad-bc$.

$$-3\left(2\times 8-3\times 6\right)+4\left(2\times 4-3\times 9\right)$$

Simplify.

$$-3\left(-2\right)+4\left(-19\right)$$

Add the terms to obtain the final result.

$$-70$$

Evaluate

$\left(\begin{matrix}0&3&4\\2&9&6\\3&4&8\end{matrix}\right)$

Invert Matrix

$\left(\begin{matrix}-\frac{24}{35}&\frac{4}{35}&\frac{9}{35}\\-\frac{1}{35}&\frac{6}{35}&-\frac{4}{35}\\\frac{19}{70}&-\frac{9}{70}&\frac{3}{35}\end{matrix}\right)$

Calculate Trace

$17$

Transpose Matrix

$\left(\begin{matrix}0&2&3\\3&9&4\\4&6&8\end{matrix}\right)$

Matrix Size

$3,3$

Reduce Matrix

$\left(\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}\right)$

Adjoint Matrix

$\left[\begin{array}{rrr}48 & -8 & -18 \\ 2 & -12 & 8 \\ -19 & 9 & -6\end{array}\right]$

To find the adjoint of a matrix, first find the cofactor matrix of the given matrix. Then find the transpose of the cofactor matrix.

Cofactor of 0

$A_{11}=\left|\begin{array}{rr}9 & 6 \\ 4 & 8\end{array}\right|=48$

Cofactor of 3

$A_{12}=-\left|\begin{array}{rr}2 & 6 \\ 3 & 8\end{array}\right|=2$

Cofactor of 4

$A_{13}=\left|\begin{array}{rr}2 & 9 \\ 3 & 4\end{array}\right|=-19$

Cofactor of 2

$A_{21}=-\left|\begin{array}{rr}3 & 4 \\ 4 & 8\end{array}\right|=-8$

Cofactor of 9

$A_{22}=\left|\begin{array}{rr}0 & 4 \\ 3 & 8\end{array}\right|=-12$

Cofactor of 6

$A_{23}=-\left|\begin{array}{rr}0 & 3 \\ 3 & 4\end{array}\right|=9$

Cofactor of 3

$A_{31}=\left|\begin{array}{rr}3 & 4 \\ 9 & 6\end{array}\right|=-18$

Cofactor of 4

$A_{32}=-\left|\begin{array}{rr}0 & 4 \\ 2 & 6\end{array}\right|=8$

Cofactor of 8

$A_{33}=\left|\begin{array}{rr}0 & 3 \\ 2 & 9\end{array}\right|=-6$

The cofactor matrix of A is [Aij]

$\left[\begin{array}{rrr}48 & 2 & -19 \\ -8 & -12 & 9 \\ -18 & 8 & -6\end{array}\right]$

The transpose of Aij

$\left[\begin{array}{rrr}48 & -8 & -18 \\ 2 & -12 & 8 \\ -19 & 9 & -6\end{array}\right]$