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M5.2 Determinants, Cofactors, and Adjoints

There are other important concepts related to matrices. These include the determinant, cofactor, and adjoint of a matrix.

Determinants

A determinant is a value associated with a square matrix. As a mathematical tool, determinants are of value in helping to solve a series of simultaneous equations.

A 2-row by 2-column $(2 \times 2)$ $(2 \times 2)$ determinant can be expressed by enclosing vertical lines around the matrix, as shown here:

| \begin{matrix} a & b \\ c & d \end{matrix} |

$| \begin{matrix} a & b \\ c & d \end{matrix} |$

Similarly, a $3 \times 3$ $3 \times 3$ determinant is indicated as

[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]

$[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]$

One common procedure for finding the determinant of a $2 \times 2$ $2 \times 2$ or $3 \times 3$ $3 \times 3$ matrix is to draw its primary and secondary diagonals. In the case of a $2 \times 2$ $2 \times 2$ determinant, the value is found by multiplying the numbers on the primary diagonal and subtracting from that product the product of the numbers on the secondary diagonal:

An image shows the formula for calculating the value of a matrix and an image shows the diagonals of a matrix.

5.1-5 Full Alternative Text

For a $3 \times 3$ $3 \times 3$ matrix, we redraw the first two columns to help visualize all diagonals and follow a similar procedure:

An image shows the calculation of the value of a 3 by 3 matrix.

5.1-6 Full Alternative Text

Let’s use this approach to find the numerical values of the following $2 \times 2$ $2 \times 2$ and $3 \times 3$ $3 \times 3$ determinants:

An image shows the calculation of the values of a 2 by 2 matrix and a 3 by 3 matrix.

5.1-7 Full Alternative Text

A set of simultaneous equations can be solved through the use of determinants by setting up a ratio of two special determinants for each unknown variable. This fairly easy procedure is best illustrated with an example.

Given the three simultaneous equations

\begin{array}{l} 2 X & + & 3 Y & + & 1 Z & = & 10 \\ 4 X & - & 1 Y & - & 2 Z & = & 8 \\ 5 X & + & 2 Y & - & 3 Z & = & 6 \end{array}

$\begin{array}{l} 2 X & + & 3 Y & + & 1 Z & = & 10 \\ 4 X & - & 1 Y & - & 2 Z & = & 8 \\ 5 X & + & 2 Y & - & 3 Z & = & 6 \end{array}$

we can structure determinants to help solve for unknown quantities $X, Y,$ $X, Y,$ and Z:

An image shows the solving for X, Y, and Z as the division of 2 3 by 3 determinants.

5.1-8 Full Alternative Text

Determining the values of X, Y, and Z now involves finding the numerical values of the four separate determinants using the method shown earlier in this module:

\begin{array}{l} X & = & \frac{Numerical value of numerator determinant}{Numerical value of denominator determinant} \\ = & \frac{128}{33} = 3.88 \\ Y & = & \frac{- 20}{33} = - 0.61 \\ Z & = & \frac{134}{33} = 4.06 \end{array}

$\begin{array}{l} X & = & \frac{Numerical value of numerator determinant}{Numerical value of denominator determinant} \\ = & \frac{128}{33} = 3.88 \\ Y & = & \frac{- 20}{33} = - 0.61 \\ Z & = & \frac{134}{33} = 4.06 \end{array}$

To verify that $X = 3.88, Y = - 0.61,$ $X = 3.88, Y = - 0.61,$ and $Z = 4.06,$ $Z = 4.06,$ we may choose any one of the original three simultaneous equations and insert these numbers. For example,

\begin{array}{l} 2 X + 3 Y + 1 Z = 10 & = & 10 \\ 2 (3.88) + 3 (- 0.61) + 1 (4.06) & = & 7.76 - 1.83 + 4.06 \\ = & 10 \end{array}

$\begin{array}{l} 2 X + 3 Y + 1 Z = 10 & = & 10 \\ 2 (3.88) + 3 (- 0.61) + 1 (4.06) & = & 7.76 - 1.83 + 4.06 \\ = & 10 \end{array}$

Matrix of Cofactors and Adjoint

Two more useful concepts in the mathematics of matrices are the matrix of cofactors and the adjoint of a matrix. A cofactor is defined as the set of numbers that remains after a given row and column have been taken out of a matrix. An adjoint is simply the transpose of the matrix of cofactors. The real value of the two concepts lies in their usefulness in forming the inverse of a matrix—something that we investigate in the next section.

To compute the matrix of cofactors for a particular matrix, we follow six steps:

Six Steps in Computing a Matrix of Cofactors

Select an element in the original matrix.
Draw a line through the row and column of the element selected. The numbers uncovered represent the cofactor for that element.
Calculate the value of the determinant of the cofactor.
Add together the location numbers of the row and column crossed out in step 2. If the sum is even, the sign of the determinant’s value (from step 3) does not change. If the sum is an odd number, change the sign of the determinant’s value.
The number just computed becomes an entry in the matrix of cofactors; it is located in the same position as the element selected in step 1.
Return to step 1 and continue until all elements in the original matrix have been replaced by their cofactor values.

Let’s compute the matrix of cofactors, and then the adjoint, for the following matrix, using Table M5.1 to help in the calculations:

5.1-9 Full Alternative Text

Table M5.1 Matrix of Cofactor Calculations

ELEMENT REMOVED	COFACTORS	DETERMINANT OF COFACTORS	VALUE OF COFACTOR
Row 1, column 1	$(\begin{matrix} 0 & 3 \\ 1 & 8 \end{matrix})$ $(\begin{matrix} 0 & 3 \\ 1 & 8 \end{matrix})$	$\| \begin{matrix} 0 & 3 \\ 1 & 8 \end{matrix} \| = - 3$ $\| \begin{matrix} 0 & 3 \\ 1 & 8 \end{matrix} \| = - 3$	$- 3$ $- 3$ (sign not changed)
Row 1, column 2	$(\begin{matrix} 2 & 3 \\ 4 & 8 \end{matrix})$ $(\begin{matrix} 2 & 3 \\ 4 & 8 \end{matrix})$	$\| \begin{matrix} 2 & 3 \\ 4 & 8 \end{matrix} \| = 4$ $\| \begin{matrix} 2 & 3 \\ 4 & 8 \end{matrix} \| = 4$	$- 4$ $- 4$ (sign changed)
Row 1, column 3	$(\begin{matrix} 2 & 0 \\ 4 & 1 \end{matrix})$ $(\begin{matrix} 2 & 0 \\ 4 & 1 \end{matrix})$	$\| \begin{matrix} 2 & 0 \\ 4 & 1 \end{matrix} \| = 2$ $\| \begin{matrix} 2 & 0 \\ 4 & 1 \end{matrix} \| = 2$	2 (sign not changed)
Row 2, column 1	$(\begin{matrix} 7 & 5 \\ 1 & 8 \end{matrix})$ $(\begin{matrix} 7 & 5 \\ 1 & 8 \end{matrix})$	$\| \begin{matrix} 7 & 5 \\ 1 & 8 \end{matrix} \| = 51$ $\| \begin{matrix} 7 & 5 \\ 1 & 8 \end{matrix} \| = 51$	$- 51$ $- 51$ (sign changed)
Row 2, column 2	$(\begin{matrix} 3 & 5 \\ 4 & 8 \end{matrix})$ $(\begin{matrix} 3 & 5 \\ 4 & 8 \end{matrix})$	$\| \begin{matrix} 3 & 5 \\ 4 & 8 \end{matrix} \| = 4$ $\| \begin{matrix} 3 & 5 \\ 4 & 8 \end{matrix} \| = 4$	4 (sign not changed)
Row 2, column 3	$(\begin{matrix} 3 & 7 \\ 4 & 1 \end{matrix})$ $(\begin{matrix} 3 & 7 \\ 4 & 1 \end{matrix})$	$\| \begin{matrix} 3 & 7 \\ 4 & 1 \end{matrix} \| = - 25$ $\| \begin{matrix} 3 & 7 \\ 4 & 1 \end{matrix} \| = - 25$	25 (sign changed)
Row 3, column 1	$(\begin{matrix} 7 & 5 \\ 0 & 3 \end{matrix})$ $(\begin{matrix} 7 & 5 \\ 0 & 3 \end{matrix})$	$\| \begin{matrix} 7 & 5 \\ 0 & 3 \end{matrix} \| = 21$ $\| \begin{matrix} 7 & 5 \\ 0 & 3 \end{matrix} \| = 21$	21 (sign not changed)
Row 3, column 2	$(\begin{matrix} 3 & 5 \\ 2 & 3 \end{matrix})$ $(\begin{matrix} 3 & 5 \\ 2 & 3 \end{matrix})$	$\| \begin{matrix} 3 & 5 \\ 2 & 3 \end{matrix} \| = - 1$ $\| \begin{matrix} 3 & 5 \\ 2 & 3 \end{matrix} \| = - 1$	1 (sign changed)
Row 3, column 3	$(\begin{matrix} 3 & 7 \\ 2 & 0 \end{matrix})$ $(\begin{matrix} 3 & 7 \\ 2 & 0 \end{matrix})$	$\| \begin{matrix} 3 & 7 \\ 2 & 0 \end{matrix} \| = - 14$ $\| \begin{matrix} 3 & 7 \\ 2 & 0 \end{matrix} \| = - 14$	$- 14$ $- 14$ (sign not changed)

Table M5.1 Full Alternative Text

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Table of Contents for M5.2 Determinants, Cofactors, and Adjoints

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M5.2 Determinants, Cofactors, and Adjoints