Determinant of a 4x4 matrix: Calculation and Solved Example

Determinant of a 4x4 matrix is a number that represents the size and orientation of the matrix. 

• It is calculated by multiplying certain elements of the matrix together in a specific order and then adding or subtracting the results based on the position of the elements in the matrix. 
• The determinant can be used to determine things such as whether the matrix is invertible (able to be turned back into its original form) or if it represents a transformation (change in position or orientation).

Key Terms: Matrix, determinant, Row, Column, Inverse, Element.

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How to Calculate the Determinant of the 4x4 Matrix?

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To calculate the determinant of a 4x4 matrix, one needs to follow these steps:

• Write down the matrix in a grid format with 4 columns and 4 rows.
• Take the first element in the first row (also called the "leading element") and multiply it by the determinant of the 3x3 matrix that is formed by deleting the row and column where the leading element is located.
• Repeat step 2 for the second element in the first row, but this time you need to multiply the determinant of the 3x3 matrix by -1.
• Repeat steps 2 and 3 for the third and fourth elements in the first row.
• Add up all the results from steps 2, 3, and 4. This will give you the determinant of the 4x4 matrix.

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Conditions for Determinant of 4x4 Matrix to be Zero

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The following are some situations where the determinant of a 44 matrix has a value of zero:

Case 1: In this instance, one of the matrix's rows is zero, making the value of the matrix's determinant 0.

\(\begin{bmatrix}7 & 3 & 1 & 5\\[0.3em]1 & 8 & 3 & 6\\[0.3em]0 & 0 & 0 & 0\\[0.3em]1 & 1 & 5 & 2\\[0.3em] \end{bmatrix}\)

Case 2: The value for the Determinant of the 4 X 4 Matrix is zero when the first and third columns of a matrix have the identical values.

\(\begin{bmatrix}2 & 3 & 2 & 5\\[0.3em]1 & 2 & 1 & 4\\[0.3em]0 & 1 & 0 & 3\\[0.3em]1 & 0 & 1 & 2\\[0.3em] \end{bmatrix}\)

Case 3: The second and third rows are equidistant from one another. Therefore, the determinant for the 4 X 4 Matrix has a value of zero.

\(\begin{bmatrix}2 & 3 & 2 & 5\\[0.3em]2 & 5 & 7 & 3\\[0.3em]4 & 10 & 14 & 6\\[0.3em]1 & 0 & 1 & 2\\[0.3em] \end{bmatrix}\)

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Important Terms 

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A few important terms related to the determinant of the matrix are– 

• Matrix: a rectangular array of numbers or variables arranged in rows and columns
• Determinant: a scalar value that represents the unique properties of a matrix
• Inverse: a matrix that, when multiplied by the original matrix, results in the identity matrix
• Linear transformation: a mathematical function that changes the position of an object in space
• Eigenvalue: a scalar value that represents the magnitude of the change in an object's position in space.

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Solved Example

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Example:- 

For example, if the matrix is:

\(\begin{bmatrix}2 & 3 & 4 & 5\\[0.3em]1 & 2 & 3 & 4\\[0.3em]0 & 1 & 2 & 3\\[0.3em]1 & 0 & 1 & 2\\[0.3em] \end{bmatrix}\)

To find the determinant, we use the formula:

\(\begin{bmatrix}2 & 3 & 4 & 5\\[0.3em]1 & 2 & 3 & 4\\[0.3em]0 & 1 & 2 & 3\\[0.3em]1 & 0 & 1 & 2\\[0.3em] \end{bmatrix}\)

= [2* (2*2 - 3*1)] - [3 * (1*3 - 4*0)] + [4 * (1*2 - 3*1)] - [5 * (0*2 - 1*1)]

= 2 * (4 – 3) - 3 * (3 – 0) + 4 * (2 – 3) - 5 * (0 – 1)

= 2 * 1 - 3 * 3 + 4 * -1 - 5 * -1

= 2 - 9 - 4 + 5

= – 6

So the determinant of this 4x4 matrix is – 6.

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Things to Remember 

• A 4x4 matrix has four rows and four columns.
• To find the determinant of a 4x4 matrix, one must use the elements of the matrix in the formula above.
• It is important to keep track of the signs (+ or -) when solving for the determinant as they change based on the elements used in the formula.
• Remember that the determinant of a 4x4 matrix can only be found if the matrix is square and has four rows and four columns.
• A determinant of 0 means the matrix is not invertible, and the system of equations has no unique solution.
• The determinant can also be used to find the area or volume of a geometric figure in multivariable calculus.

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