Determinant of a Matrix

Pronunciation: /dɪˈtɜr mə nənt/ Explain

A determinant of a square matrix is a value calculated from the elements of the matrix.[1] A determinant is defined only for a square matrix. The determinant of matrix A is denoted |A| or det(A).

A determinant is calculated by multiplying the diagonals of the matrix, then adding or subtracting the products of the diagonals. For the 2x2 matrix

B=[b11,b12,b21,b22] square matrix
multiply the diagonal from the upper left to lower right first (b11·b22).
B=[b11,b12,b21,b22] square matrix with b11 b22 highlighted.
Then multiply the diagonal from the upper right to the lower left (b12·b21).
B=[b11,b12,b21,b22] square matrix with b12 b21 highlighted.
Subtract the second product from the first (b11·b22 - b12·b21). This is the determinant of the matrix.

Example 1

Matrix B first row 2, -1, second row 3, -2
determinant of B = 2*(-2)-(-1*3) = -4-(-3) = -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 a11·a22·a33, then a12·a23·a31, then a13·a21·a32. The product of these diagonals are added together.

The diagonals from upper right to bottom left are calculated. The numbers multiplied are a13·a22·a31, then a12·a21·a33, then a11·a23·a32. These products are subtracted from the previous sum.

The function for the determinant of a 3x3 matrix is |A| = a11·a22·a33 + a12·a23·a31 + a13·a21·a32 - a13·a22·a31 - a12·a21·a33 - a11·a23·a32.

A 3x3 matrix with the diagonals highlighted showing how to find the determinant.
Figure 1: The determinant of a 3x3 matrix

Example 2

3x3 matrix, first row 3, -2, 0; second row 1, 4, -3; third row -1, 2, 2.
calculations for the determinant of a 3x3 matrix

References

  1. determinant. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 8/6/2018. http://www.merriam-webster.com/dictionary/determinant.
  2. Ferrar, W. L.. Algebra: A Text-Book of Determinants, Matrices, and Algebraic Forms. 2nd edition. www.archive.org. NULl. Oxford University Press. 1960. Last Accessed 8/6/2018. http://www.archive.org/stream/algebra032104mbp#page/n10/mode/1up/search/determinant. Buy the book
  3. Bôcher, Maxime. Introduction to Higher Algebra. pp 20-26. www.archive.org. The Macmillan Company. July 1947. Last Accessed 8/6/2018. http://www.archive.org/stream/cu31924002936536#page/n35/mode/1up/search/determinant. Buy the book
  4. Ferrar, W. L.. Algebra. 2nd edition. pp 5-66. www.archive.org. Oxford University Press. 1960. Last Accessed 8/6/2018. 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. http://www.allmathwords.org/en/m/matrix.html.

Cite this article as:

McAdams, David E. Determinant of a Matrix. 7/4/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/determinant.html.

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Revision History

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.)

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