Studdy AI Logo
Math

QuestionFind the inverse of the matrix A=[111011001]A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}.

Studdy Solution

STEP 1

Assumptions1. Matrix A is a3x3 matrix with elements as given. . Matrix A is invertible, i.e., it has a non-zero determinant.

STEP 2

First, we need to find the determinant of matrix A. The determinant of ax matrix is calculated as followsdet(A)=a(ei−fh)−b(di−fg)+c(dh−eg)\text{det}(A) = a(ei−fh)−b(di−fg)+c(dh−eg) where a, b, c, d, e, f, g, h, i are the elements of the matrix. In this case, a, e, i =1, and all other elements are0.

STEP 3

Now, plug in the values to calculate the determinant.
det(A)=1(1∗1−0∗0)−0(0∗1−1∗0)+0(0∗0−1∗0)\text{det}(A) =1(1*1−0*0)−0(0*1−1*0)+0(0*0−1*0)

STEP 4

Calculate the determinant.
det(A)=1(1)−0+0=1\text{det}(A) =1(1)−0+0 =1

STEP 5

Now that we have the determinant and it is non-zero, we can find the adjoint of matrix A. The adjoint of a3x3 matrix is the transpose of the cofactor matrix.

STEP 6

The cofactor matrix is obtained by calculating the determinant of the2x2 sub-matrices that can be formed by removing the row and column of each element. In this case, the cofactor matrix isCofactor(A)=[det[1101]−det[0101]det[0100]−det[1101]det[1101]−det[1100]det[1101]−det[1100]det[1100]]Cofactor(A) = \left[\begin{array}{lll} \text{det}\left[\begin{array}{ll} 1 &1 \\ 0 &1\end{array}\right] & -\text{det}\left[\begin{array}{ll} 0 &1 \\ 0 &1\end{array}\right] & \text{det}\left[\begin{array}{ll} 0 &1 \\ 0 &0\end{array}\right] \\ -\text{det}\left[\begin{array}{ll} 1 &1 \\ 0 &1\end{array}\right] & \text{det}\left[\begin{array}{ll} 1 &1 \\ 0 &1\end{array}\right] & -\text{det}\left[\begin{array}{ll} 1 &1 \\ 0 &0\end{array}\right] \\ \text{det}\left[\begin{array}{ll} 1 &1 \\ 0 &1\end{array}\right] & -\text{det}\left[\begin{array}{ll} 1 &1 \\ 0 &0\end{array}\right] & \text{det}\left[\begin{array}{ll} 1 &1 \\ 0 &0\end{array}\right] \end{array}\right]

STEP 7

Calculate the cofactor matrix.
Cofactor(A)=[1−10−11−11−11]Cofactor(A) = \left[\begin{array}{lll} 1 & -1 &0 \\ -1 &1 & -1 \\ 1 & -1 &1\end{array}\right]

STEP 8

Now, take the transpose of the cofactor matrix to get the adjoint of A.
Adjoint(A)=[1−11−11−10−11]Adjoint(A) = \left[\begin{array}{lll} 1 & -1 &1 \\ -1 &1 & -1 \\ 0 & -1 &1\end{array}\right]

STEP 9

Finally, the inverse of matrix A is obtained by dividing the adjoint of A by the determinant of A.
A−=det(A)×Adjoint(A)A^{-} = \frac{}{\text{det}(A)} \times Adjoint(A)

STEP 10

Plug in the values for the determinant and the adjoint to calculate the inverse of A.
A−=×[−−−0−]A^{-} = \frac{}{} \times \left[\begin{array}{lll} & - & \\ - & & - \\ 0 & - &\end{array}\right]

STEP 11

Calculate the inverse of A.
A−=[−−−0−]A^{-} = \left[\begin{array}{lll} & - & \\ - & & - \\ 0 & - &\end{array}\right] So, the inverse of matrix A isA−=[−−−0−]A^{-} = \left[\begin{array}{lll} & - & \\ - & & - \\ 0 & - &\end{array}\right]

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord