Determinant of a Matrix
Pronunciation: /dɪˈtɜr.mə.nənt ʌv ə ˈmeɪ.trɪks/ Explain
A determinant of a square matrix is a value calculated
from the
elements
of the
matrix.^{[3]}
A determinant is
defined
only for a
square matrix.
The determinant of matrix A is denoted
A or
det(A).
How to Calculate the Determinant of a Matrix
A determinant is calculated by multiplying the
diagonals of the matrix,
then adding or subtracting the products of the diagonals. For the 2x2 matrix
multiply the diagonal from the upper left to lower right first
(
b_{11} · b_{22}).
Then multiply the diagonal from the upper right to the lower left
(
b_{12} · b_{21}).
Subtract the second product from the first
(
b_{11} · b_{22} 
b_{12} · b_{21}).
This is the determinant of the matrix.
Example 1
Determinant of a 3x3 matrix
Figure 1 shows how to find the determinant of a 3x3 matrix.. The first diagonals
go from top left to bottom right. The numbers multiplied are
a_{11} · a_{22} · a_{33},
then
a_{12}·a_{23}·a_{31},
then
a_{13}·a_{21}·a_{32}.
The product of these diagonals are added together.
The diagonals from upper right to bottom left are calculated. The numbers multiplied are
a_{13}·a_{22}·a_{31},
then
a_{12}·a_{21}·a_{33},
then
a_{11}·a_{23}·a_{32}.
These products are subtracted from the previous sum.
The function for the determinant of a 3x3 matrix is
A = a_{11}·a_{22}·a_{33} +
a_{12}·a_{23}·a_{31} +
a_{13}·a_{21}·a_{32} 
a_{13}·a_{22}·a_{31} 
a_{12}·a_{21}·a_{33} 
a_{11}·a_{23}·a_{32}.

Figure 1: The determinant of a 3x3 matrix 
Example 2
References
 McAdams, David E.. All Math Words Dictionary, determinant. 2nd Classroom edition 201501084799968. pg 58. Life is a Story Problem LLC. January 8, 2015. Buy the book
 determinant. merriamwebster.com. Encyclopedia Britannica. MerriamWebster. Last Accessed 7/3/2018. http://www.merriamwebster.com/dictionary/determinant. Buy the book
 Ferrar, W. L.. Algebra: A TextBook of Determinants, Matrices, and Algebraic Forms. 2nd edition. www.archive.org. NULl. Oxford University Press. 1960. Last Accessed 7/3/2018. http://www.archive.org/stream/algebra032104mbp#page/n10/mode/1up/search/determinant. Buy the book
 Bôcher, Maxime. Introduction to Higher Algebra. pp 2026. www.archive.org. The Macmillan Company. July 1947. Last Accessed 7/3/2018. http://www.archive.org/stream/cu31924002936536#page/n35/mode/1up/search/determinant. Buy the book
 Ferrar, W. L.. Algebra. 2nd edition. pp 566. www.archive.org. Oxford University Press. 1960. Last Accessed 7/3/2018. http://www.archive.org/stream/algebra032104mbp#page/n14/mode/1up/. Buy the book
More Information
 McAdams, David E.. Matrix. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/12/2009. https://www.allmathwords.org/en/m/matrix.html.
Cite this article as:
McAdams, David E. Determinant of a Matrix. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/d/determinant.html.
Image Credits
Revision History
4/20/2019: Updated equations and expressions to the new format (
McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
7/4/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (
McAdams, David E.)
1/23/2010: Added "References". (
McAdams, David E.)
1/9/2009: Added Laplace Expansion to 'More Information'. (
McAdams, David E.)
12/27/2008: Changed '3xe matrix' to '3x3 matrix'. (
McAdams, David E.)
12/3/2008: Initial version. (
McAdams, David E.)