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Inverse Of 4x4 Matrix Example Pdf Download


Inverse of 4x4 Matrix Example PDF Download




A 4x4 matrix is a square matrix that has four rows and four columns. The inverse of a 4x4 matrix, if it exists, is another 4x4 matrix that satisfies the following property:


$$A^-1A = AA^-1 = I$$ where $A$ is the original matrix, $A^-1$ is the inverse matrix, and $I$ is the 4x4 identity matrix. The identity matrix has ones on the diagonal and zeros elsewhere.


Finding the inverse of a 4x4 matrix can be done by using various methods, such as Gaussian elimination, cofactor expansion, or adjugate formula. In this article, we will focus on the Gaussian elimination method, which involves augmenting the original matrix with the identity matrix and performing row operations to reduce the augmented matrix to reduced row echelon form. The reduced row echelon form of a matrix has ones on the diagonal, zeros below and above the diagonal, and zeros at the bottom. If the original matrix is invertible, then the reduced row echelon form of the augmented matrix will have the identity matrix on the left and the inverse matrix on the right.


Example




Let us consider the following 4x4 matrix:


$$A = \beginbmatrix 1 & 2 & -1 & 0 \\ 0 & 3 & 2 & -1 \\ 2 & -1 & 0 & -2 \\ -3 & 0 & 1 & 1 \endbmatrix$$ To find its inverse, we augment it with the 4x4 identity matrix:


$$[AI] = \beginbmatrix 1 & 2 & -1 & 0 & & 1 & 0 & 0 & 0 \\ 0 & 3 & 2 & -1 & & 0 & 1 & 0 & 0 \\ 2 & -1 & 0 & -2 & & 0 & 0 & 1 & 0 \\ -3 & 0 & 1 & 1 & & 0 & 0 & 0 & 1 \endbmatrix$$ We then perform row operations to reduce the augmented matrix to reduced row echelon form. The steps are shown below:


$$\beginalign* &\beginbmatrix 1&2&-1&0&&1&0&0&0\\ 0&3&2&-1&&0&1&0&0\\ 2&-1&0&-2&&0&0&1&0\\ -3&0&1&1&&0&0&0&1 \endbmatrix\\ \xrightarrowR_3-2R_1&\beginbmatrix 1&2&-1&0&&1&0&0&0\\ 0&3&2&-1&&0&1&0&0\\ 0&-5&2&-2&&-2&0&1&0\\ -3&0&1&1&&0&0&0&1 \endbmatrix\\ \xrightarrowR_4+3R_1&\beginbmatrix 1&2& -1&


0&&3&0&0\\ 0&3&2&-1&&0&1&0&0\\ 0&-5&2&-2&&-2&0&1&0\\ 0&6&-2&3&&3&0&0&1 \endbmatrix\\ \xrightarrowR_2/3&\beginbmatrix 1&2& -1& 0& & 3& 0& 0& 0\\ 0& 1& 2/3& -1/3& & 0& 1/3& 0& 0\\ 0& -5& 2& -2& & -2& 0& 1& 0\\ 0& 6& -2& 3& & 3& 0& 0& 1 \endbmatrix\\ \xrightarrowR_3+5R_2&\beginbmatrix 1&2& -1& 0& & 3& 0& 0& 0\\ 0& 1& 2/3& -1/3& & 0& 1/3& 0& 0\\ 0& 0& 5/3& -7/3& & -2/3 & 5/3 & 1 & 0\\ 0 & 6 & -2 & 3 & & 3 & 0 & 0 & 1 \endbmatrix\\ \xrightarrowR_4-6R_2&\beginbmatrix 1 & 2 & -1 & 0 & & 3 & 0 & 0 & 0 \\ 0 & 1 & 2/3 & -1/3 & & 0 & 1/3 & 0 & 0 \\ 0 & 0 & 5/3 & -7/3 & & -2/3 & 5/3 & 1 & 0 \\ 0 & 0 & -6 & 4 & & 3 & -2 & 0 & 1 \endbmatrix\\ \xrightarrowR_4+18/5R_3&\beginbmatrix 1 & 2 & -1 & 0 & & 3 & 0 & 0 & 0 \\ 0 & 1 & 2/3 & -1/3 & & 0 & 1/3 & 0 & 0 \\ 0 & 0 & 5/3 & -7/3 & & -2/3 & 5/3 & 1 & 0 \\


0 & 0 & 0 & -1/5 & & 3/5 & 1/5 & 1/5 & 1 \\ \endbmatrix\\ \xrightarrowR_4*(-5)&\beginbmatrix 1 & 2 & -1 & 0 & & 3 & 0 & 0 & 0 \\ 0 & 1 & 2/3 & -1/3 & & 0 & 1/3 & 0 & 0 \\ 0 & 0 & 5/3 & -7/3 & & -2/3 & 5/3 & 1 & 0 \\ 0 & 0 & 0 & 1 & & -3 & -1 & -1 & -5 \endbmatrix\\ \xrightarrowR_3+7/5R_4&\beginbmatrix 1&2&-1&0&&3&0&0&0\\ 0&1&2/3&-1/3&&0&1/3&0&0\\ 0&0&5/3&0&&-9/5&2&-2/5&-7\\ 0&0&0&1&&-3&-1&-1&-5 \endbmatrix\\ \xrightarrowR_2+R_4/3&\beginbmatrix 1&2& -1&


0&&-1&-1/3&-1/3\\ 0&0&5/3&0&&-9/5&2&-2/5&-7\\ 0&0&0&1&&-3&-1&-1&-5 \endbmatrix\\ \xrightarrowR_1+R_4&\beginbmatrix 1 & 2 & -1 & 0 & & 0 & -1 & -1 & -5 \\ 0 & 1 & 2/3 & 0 & & -1 & 0 & -1/3 & -5/3 \\ 0 & 0 & 5/3 & 0 & & -9/5 & 2 & -2/5 & -7 \\ 0 & 0 & 0 & 1 & & -3 & -1 & -1 & -5 \endbmatrix\\ \xrightarrowR_1-2R_2&\beginbmatrix 1 & 0 & -7/3 & 0 & & 2 & -1 & -1/3 & -5/3 \\ 0 & 1 & 2/3 & 0 & & -1 & 0 & -1/3 & -5/3 \\ 0 & 0 & 5/3 & 0 & & -9/5 & 2 & -2/5 & -7 \\ 0 &


0 & 0 & 1 & 0 & & -3 & -1 & -1 & -5 \\ \endbmatrix\\ \xrightarrowR_3*(3/5)&\beginbmatrix 1 & 0 & -7/3 & 0 & & 2 & -1 & -1/3 & -5/3 \\ 0 & 1 & 2/3 & 0 & & -1 & 0 & -1/3 & -5/3 \\ 0 & 0 & 1 & 0 & & -9/25 & 6/5 & -6/25 & -21/5 \\ 0 &


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