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Inverses and the Adjoint Matrix

We now use Cramer’s rule to develop an explicit formula for the inverse $A^{- 1}$ $A^{- 1}$ of the invertible matrix A. First, we need to rewrite Cramer’s rule more concisely. Expansion of the determinant in the numerator in Eq. (24) along its i th column yields

x_{i} = \frac{1}{| A |} (b_{1} A_{1 i} + b_{2} A_{2 i} + \dots + b_{n} A_{n i}),

$x_{i} = \frac{1}{| A |} (b_{1} A_{1 i} + b_{2} A_{2 i} + \dots + b_{n} A_{n i}),$ (25)

because the cofactor of $b_{p}$ $b_{p}$ is simply the cofactor $A_{p i}$ $A_{p i}$ of $a_{p i}$ $a_{p i}$ in $| A | .$ $| A | .$ The formula in Eq. (25) gives the solution vector

x = [\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}] = \frac{1}{| A |} [\begin{array}{c} b_{1} A_{11} & + & b_{2} A_{21} & + & \dots & + & b_{n} A_{n 1} \\ b_{1} A_{12} & + & b_{2} A_{22} & + & \dots & + & b_{n} A_{n 2} \\ ⋮ \\ b_{1} A_{1 n} & + & b_{2} A_{2 n} & + & \dots & + & b_{n} A_{n n} \end{array}] .

$x = [\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}] = \frac{1}{| A |} [\begin{array}{c} b_{1} A_{11} & + & b_{2} A_{21} & + & \dots & + & b_{n} A_{n 1} \\ b_{1} A_{12} & + & b_{2} A_{22} & + & \dots & + & b_{n} A_{n 2} \\ ⋮ \\ b_{1} A_{1 n} & + & b_{2} A_{2 n} & + & \dots & + & b_{n} A_{n n} \end{array}] .$

Then the definition of matrix multiplication yields

x = \frac{1}{| A |} [\begin{array}{c} A_{11} & A_{21} & \dots & A_{n 1} \\ A_{12} & A_{22} & \dots & A_{n 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ A_{1 n} & A_{2 n} & \dots & A_{n n} \end{array}] [\begin{array}{c} b_{1} \\ b_{2} \\ ⋮ \\ b_{n} \end{array}]

$x = \frac{1}{| A |} [\begin{array}{c} A_{11} & A_{21} & \dots & A_{n 1} \\ A_{12} & A_{22} & \dots & A_{n 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ A_{1 n} & A_{2 n} & \dots & A_{n n} \end{array}] [\begin{array}{c} b_{1} \\ b_{2} \\ ⋮ \\ b_{n} \end{array}]$ (26)

for the solution x of $A x = b .$ $A x = b .$

Observe that the double subscripts in (26) are reversed from their usual order; the element in the ith row and jth column is $A_{j i}$ $A_{j i}$ (rather than $A_{i j}$ $A_{i j}$ ). We therefore see in (26) the transpose of the cofactor matrix $[A_{i j}]$ $[A_{i j}]$ of then $n \times n$ $n \times n$ matrix A. The transpose of the cofactor matrix of A is called the adjoint matrix of A and is denoted by

adj A = {[A_{i j}]}^{T} = [A_{i j}] .

$adj A = {[A_{i j}]}^{T} = [A_{i j}] .$ (27)

With the aid of this notation, Cramer’s rule as expressed in Eq. (26) can be written in the especially simple form

x = \frac{[A_{j i}] b}{| A |} = \frac{(adj A) b}{| A |} .

$x = \frac{[A_{j i}] b}{| A |} = \frac{(adj A) b}{| A |} .$ (28)

The fact that the formula in (28) gives the unique solution x of $A x = b$ $A x = b$ implies that

A \frac{(adj A) b}{| A |} = b

$A \frac{(adj A) b}{| A |} = b$ (29)

for every n-vector b. If we write

C = \frac{adj A}{| A |}

$C = \frac{adj A}{| A |}$ (30)

for brevity, then

A C b = b

$A C b = b$ (31)

for every n-vector b. From this it follows (one column at a time, as we use Fact 2 in Section 3.5) that

A C B = B

$A C B = B$ (32)

for every matrix B having n rows. In particular, with $B = I$ $B = I$ (the $n \times n$ $n \times n$ identity matrix), we see that

A C = I .

$A C = I .$ (33)

Therefore, we have discovered that the matrix C as defined in Eq. (30) is the inverse matrix $A^{- 1}$ $A^{- 1}$ of A.

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Table of Contents for Inverses and the Adjoint Matrix

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Table of Contents for
Inverses and the Adjoint Matrix