Matrices and Determinants Calculator

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Example

Created on 2024-06-20Asked by Abigail Rodriguez (Solvelet student)

Find the determinant of the matrix

A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}.

Solution

To find the determinant of the matrix

A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix},

we use the cofactor expansion along the first row: 1. The determinant of

A

is:

\det(A) = 1 \cdot \begin{vmatrix} 1 & 4 \\ 6 & 0 \end{vmatrix} - 2 \cdot \begin{vmatrix} 0 & 4 \\ 5 & 0 \end{vmatrix} + 3 \cdot \begin{vmatrix} 0 & 1 \\ 5 & 6 \end{vmatrix}

2. Calculate the 2x2 determinants (minors):

\begin{vmatrix} 1 & 4 \\ 6 & 0 \end{vmatrix} = (1)(0) - (4)(6) = -24

\begin{vmatrix} 0 & 4 \\ 5 & 0 \end{vmatrix} = (0)(0) - (4)(5) = -20

\begin{vmatrix} 0 & 1 \\ 5 & 6 \end{vmatrix} = (0)(6) - (1)(5) = -5

3. Substitute these values into the cofactor expansion:

\det(A) = 1 \cdot (-24) - 2 \cdot (-20) + 3 \cdot (-5)

4. Simplify the expression:

\det(A) = -24 + 40 - 15 = 1

Therefore, the determinant of the matrix

A

is:

\det(A) = 1

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Jacob Gonzalez on Solvelet

1. Find the inverse of the matrix

A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}

.2. Solve the system of equations represented by the matrix equation

AX = B

, where

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

and

B = \begin{bmatrix} 5 \\ 7 \end{bmatrix}

DefinitionArrays in mathematics, rectilinear arrays of numbers, which is the matrix, while a determinant is scalar associated with a square matrix. The determinant also conveys crucial qualities of the matrix regarding whether it is invertible. Now, the determinant of a 2x2 matrix [ ac bd ] = ad−bc For example: If the matrix is given (1324), then the determinant is 1∗4−2∗3=−2

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