Solve The Matrix Equation Calculator
Solution example
A =
| 000 | 71 | 8 | 8 | 2 | 000 |
| 7 | 8 | five | 2 | ||
| 2 | five | 8 | 7 | ||
| iv | 5 | five | 2 |
B =
| 000 | 1 | 2 | 3 | 4 | 000 |
| v | vi | 7 | 8 | ||
| 9 | 10 | 11 | 12 | ||
| 13 | 14 | 15 | 16 |
Since A * 10 = B, information technology follows that X = A<sup>-1</sup> * B;
To observe the roots of the equation, we summate the inverse matrix of matrix A and multiply it by matrix B;
We reduce the matrix to a triangular form, so the product of the elements of the main diagonal will give the states the determinant;
det(A) =
| 000 | 71 | 8 | 8 | 2 | 000 |
| 7 | 8 | 5 | ii | ||
| 2 | 5 | eight | 7 | ||
| 4 | 5 | five | ii |
=
from 2 row we subtruct 1 row, multiplied by 0.09859;
from 3 row nosotros subtruct 1 row, multiplied by 0.02817;
from iv row nosotros subtruct ane row, multiplied by 0.05634;
=
| 000 | 71 | 8 | viii | two | 000 |
| 0 | 7.21128 | 4.21128 | 1.80282 | ||
| 0 | 4.77464 | vii.77464 | 6.94366 | ||
| 0 | 4.54928 | 4.54928 | one.88732 |
=
from 3 row nosotros subtruct ii row, multiplied by 0.66211;
from 4 row we subtruct 2 row, multiplied by 0.63086;
=
| 000 | 71 | 8 | 8 | 2 | 000 |
| 0 | 7.21128 | 4.21128 | 1.80282 | ||
| 0 | 0 | 4.98631 | 5.74999 | ||
| 0 | 0 | 1.89255 | 0.74999 |
=
from four row we subtruct 3 row, multiplied by 0.37955;
=
| 000 | 71 | 8 | viii | 2 | 000 |
| 0 | 7.21128 | 4.21128 | 1.80282 | ||
| 0 | 0 | iv.98631 | 5.74999 | ||
| 0 | 0 | 0 | -1.43242 |
=
det(A) = 71 * 7.21128 * 4.98631 * -one.43242 = -3657;
Matrix of minors:
Thou =
| 000 | -57 | -57 | -three | 21 | 000 |
| xviii | -1201 | -1603 | -969 | ||
| -18 | -18 | 384 | 969 | ||
| -138 | 1081 | 2944 | 2553 |
Calculation of elements of the matrix: 
M11 =
| 000 | 71 | eight | eight | two | 000 |
| vii | 8 | 5 | ii | ||
| two | five | eight | 7 | ||
| iv | five | v | 2 |
=
| 000 | 8 | 5 | two | 000 |
| five | viii | 7 | ||
| v | five | two |
= -57
M12 =
| 000 | 71 | eight | 8 | two | 000 |
| 7 | eight | 5 | two | ||
| ii | 5 | eight | vii | ||
| 4 | 5 | 5 | 2 |
=
| 000 | vii | 5 | 2 | 000 |
| ii | 8 | 7 | ||
| 4 | 5 | ii |
= -57
M13 =
| 000 | 71 | 8 | 8 | 2 | 000 |
| 7 | 8 | 5 | 2 | ||
| two | five | 8 | 7 | ||
| 4 | 5 | 5 | 2 |
=
| 000 | 7 | 8 | ii | 000 |
| ii | 5 | seven | ||
| 4 | five | 2 |
= -3
Thousand14 =
| 000 | 71 | 8 | 8 | 2 | 000 |
| 7 | eight | 5 | 2 | ||
| 2 | 5 | 8 | seven | ||
| 4 | 5 | 5 | 2 |
=
| 000 | 7 | 8 | 5 | 000 |
| 2 | 5 | viii | ||
| 4 | v | v |
= 21
M21 =
| 000 | 71 | 8 | 8 | 2 | 000 |
| 7 | eight | five | two | ||
| 2 | 5 | viii | 7 | ||
| 4 | 5 | 5 | ii |
=
| 000 | 8 | viii | 2 | 000 |
| 5 | 8 | 7 | ||
| 5 | 5 | 2 |
= 18
Grand22 =
| 000 | 71 | viii | viii | two | 000 |
| 7 | 8 | 5 | 2 | ||
| ii | v | eight | 7 | ||
| 4 | 5 | 5 | 2 |
=
| 000 | 71 | 8 | 2 | 000 |
| 2 | eight | 7 | ||
| 4 | v | 2 |
= -1201
K23 =
| 000 | 71 | 8 | 8 | ii | 000 |
| 7 | 8 | 5 | 2 | ||
| ii | 5 | 8 | seven | ||
| 4 | 5 | 5 | 2 |
=
| 000 | 71 | eight | ii | 000 |
| two | 5 | vii | ||
| four | 5 | 2 |
= -1603
M24 =
| 000 | 71 | 8 | eight | two | 000 |
| 7 | eight | v | 2 | ||
| 2 | 5 | 8 | 7 | ||
| iv | 5 | five | two |
=
| 000 | 71 | 8 | 8 | 000 |
| 2 | 5 | 8 | ||
| 4 | v | 5 |
= -969
M31 =
| 000 | 71 | eight | 8 | 2 | 000 |
| seven | viii | 5 | 2 | ||
| 2 | 5 | viii | 7 | ||
| four | five | five | 2 |
=
| 000 | viii | 8 | 2 | 000 |
| 8 | v | 2 | ||
| 5 | 5 | 2 |
= -xviii
Thousand32 =
| 000 | 71 | 8 | 8 | 2 | 000 |
| 7 | viii | five | ii | ||
| two | v | viii | 7 | ||
| 4 | 5 | 5 | ii |
=
| 000 | 71 | 8 | two | 000 |
| 7 | 5 | ii | ||
| four | 5 | 2 |
= -xviii
100033 =
| 000 | 71 | eight | 8 | ii | 000 |
| 7 | viii | 5 | two | ||
| two | 5 | 8 | 7 | ||
| iv | 5 | 5 | 2 |
=
| 000 | 71 | 8 | two | 000 |
| vii | 8 | 2 | ||
| 4 | 5 | 2 |
= 384
Thousand34 =
| 000 | 71 | eight | 8 | ii | 000 |
| 7 | 8 | 5 | ii | ||
| two | 5 | 8 | 7 | ||
| iv | v | 5 | 2 |
=
| 000 | 71 | eight | 8 | 000 |
| 7 | 8 | 5 | ||
| 4 | 5 | v |
= 969
G41 =
| 000 | 71 | 8 | 8 | 2 | 000 |
| 7 | 8 | 5 | ii | ||
| 2 | 5 | 8 | seven | ||
| 4 | 5 | v | ii |
=
| 000 | eight | eight | 2 | 000 |
| 8 | 5 | 2 | ||
| five | viii | 7 |
= -138
Thousand42 =
| 000 | 71 | 8 | 8 | 2 | 000 |
| 7 | 8 | 5 | 2 | ||
| 2 | 5 | 8 | 7 | ||
| 4 | 5 | 5 | ii |
=
| 000 | 71 | 8 | 2 | 000 |
| seven | five | ii | ||
| 2 | eight | 7 |
= 1081
M43 =
| 000 | 71 | viii | 8 | 2 | 000 |
| seven | 8 | 5 | two | ||
| 2 | 5 | eight | seven | ||
| 4 | 5 | 5 | 2 |
=
| 000 | 71 | 8 | ii | 000 |
| 7 | viii | two | ||
| 2 | 5 | 7 |
= 2944
M44 =
| 000 | 71 | 8 | 8 | 2 | 000 |
| 7 | viii | five | 2 | ||
| 2 | 5 | viii | 7 | ||
| 4 | five | 5 | 2 |
=
| 000 | 71 | viii | 8 | 000 |
| 7 | 8 | 5 | ||
| 2 | five | viii |
= 2553
Matrix of algebraic complements:
A =
| 000 | -57 | 57 | -iii | -21 | 000 |
| -eighteen | -1201 | 1603 | -969 | ||
| -18 | eighteen | 384 | -969 | ||
| 138 | 1081 | -2944 | 2553 |
Adding of elements of the matrix:
A11 = Mxi * (-1)1+ane = -57 * (-1)1+1 = -57
A12 = M12 * (-i)1+two = -57 * (-1)ane+2 = 57
Axiii = Mthirteen * (-1)ane+three = -3 * (-1)1+3 = -3
Afourteen = Mfourteen * (-1)1+4 = 21 * (-1)1+four = -21
A21 = M21 * (-i)2+1 = 18 * (-ane)2+1 = -eighteen
A22 = 100022 * (-1)2+2 = -1201 * (-1)two+two = -1201
A23 = M23 * (-1)2+3 = -1603 * (-1)2+3 = 1603
A24 = M24 * (-ane)2+four = -969 * (-1)two+4 = -969
A31 = M31 * (-1)iii+1 = -xviii * (-one)three+1 = -18
A32 = M32 * (-1)3+2 = -xviii * (-1)iii+2 = 18
A33 = M33 * (-ane)3+3 = 384 * (-one)3+three = 384
A34 = M34 * (-i)3+iv = 969 * (-1)3+4 = -969
A41 = M41 * (-one)4+1 = -138 * (-ane)4+1 = 138
A42 = One thousand42 * (-i)four+2 = 1081 * (-one)4+2 = 1081
A43 = G43 * (-ane)4+three = 2944 * (-ane)4+3 = -2944
A44 = M44 * (-1)4+4 = 2553 * (-1)4+iv = 2553
Transposed matrix of algebraic complements:
AT =
| 000 | -57 | -18 | -18 | 138 | 000 |
| 57 | -1201 | 18 | 1081 | ||
| -3 | 1603 | 384 | -2944 | ||
| -21 | -969 | -969 | 2553 |
Changed matrix:
A-1 = AT * (ane/d) =
| 000 | 0.01559 | 0.00492 | 0.00492 | -0.03774 | 000 |
| -0.01559 | 0.32841 | -0.00492 | -0.2956 | ||
| 0.00082 | -0.43834 | -0.10501 | 0.80504 | ||
| 0.00574 | 0.26497 | 0.26497 | -0.69812 |
Adding of elements of the matrix:
A-1 11 = AT 11 * (i/d) = -57 * (i/-3656.96126) = 0.01559
A-one 12 = AT 12 * (1/d) = -18 * (1/-3656.96126) = 0.00492
A-1 13 = AT 13 * (one/d) = -xviii * (1/-3656.96126) = 0.00492
A-one fourteen = AT 14 * (1/d) = 138 * (1/-3656.96126) = -0.03774
A-i 21 = AT 21 * (1/d) = 57 * (1/-3656.96126) = -0.01559
A-1 22 = AT 22 * (1/d) = -1201 * (1/-3656.96126) = 0.32841
A-1 23 = AT 23 * (ane/d) = 18 * (i/-3656.96126) = -0.00492
A-1 24 = AT 24 * (1/d) = 1081 * (i/-3656.96126) = -0.2956
A-1 31 = AT 31 * (ane/d) = -iii * (one/-3656.96126) = 0.00082
A-1 32 = AT 32 * (1/d) = 1603 * (1/-3656.96126) = -0.43834
A-i 33 = AT 33 * (1/d) = 384 * (1/-3656.96126) = -0.10501
A-1 34 = AT 34 * (1/d) = -2944 * (1/-3656.96126) = 0.80504
A-1 41 = AT 41 * (1/d) = -21 * (1/-3656.96126) = 0.00574
A-1 42 = AT 42 * (1/d) = -969 * (1/-3656.96126) = 0.26497
A-1 43 = AT 43 * (one/d) = -969 * (1/-3656.96126) = 0.26497
A-1 44 = AT 44 * (1/d) = 2553 * (one/-3656.96126) = -0.69812
A-1 * B = 10 =
| 000 | 0.01559 | 0.00492 | 0.00492 | -0.03774 | 000 |
| -0.01559 | 0.32841 | -0.00492 | -0.2956 | ||
| 0.00082 | -0.43834 | -0.10501 | 0.80504 | ||
| 0.00574 | 0.26497 | 0.26497 | -0.69812 |
*
| 000 | 1 | two | 3 | 4 | 000 |
| v | vi | 7 | 8 | ||
| ix | x | 11 | 12 | ||
| 13 | 14 | 15 | 16 |
=
| 000 | -0.40095 | -0.4131 | -0.42525 | -0.4374 | 000 |
| -2.23209 | -two.21994 | -2.20779 | -ii.19564 | ||
| vii.23708 | 7.49628 | 7.75548 | eight.01468 | ||
| -v.29254 | -5.45292 | -v.6133 | -5.77368 |
Adding of elements of the matrix:
x11 = a11 * bxi + a12 * b21 + axiii * b31 + axiv * b41 = 0.01539 * 1 + 0.00486 * 5 + 0.00486 * 9 + -0.03726 * 13 = -0.40095; x12 = a11 * b12 + a12 * b22 + axiii * b32 + afourteen * b42 = 0.01539 * 2 + 0.00486 * 6 + 0.00486 * 10 + -0.03726 * xiv = -0.4131; xxiii = axi * bxiii + a12 * b23 + athirteen * b33 + a14 * b43 = 0.01539 * iii + 0.00486 * 7 + 0.00486 * 11 + -0.03726 * 15 = -0.42525; xxiv = a11 * b14 + a12 * b24 + a13 * b34 + a14 * b44 = 0.01539 * iv + 0.00486 * viii + 0.00486 * 12 + -0.03726 * 16 = -0.4374; x21 = a21 * b11 + a22 * b21 + a23 * b31 + a24 * b41 = -0.01539 * ane + 0.32427 * five + -0.00486 * 9 + -0.29187 * 13 = -ii.23209; x22 = a21 * b12 + a22 * b22 + a23 * b32 + a24 * b42 = -0.01539 * ii + 0.32427 * 6 + -0.00486 * x + -0.29187 * 14 = -2.21994; x23 = a21 * bthirteen + a22 * b23 + a23 * b33 + a24 * b43 = -0.01539 * three + 0.32427 * 7 + -0.00486 * eleven + -0.29187 * 15 = -ii.20779; 1024 = a21 * bxiv + a22 * b24 + a23 * b34 + a24 * b44 = -0.01539 * iv + 0.32427 * 8 + -0.00486 * 12 + -0.29187 * xvi = -2.19564; 1031 = a31 * b11 + a32 * b21 + a33 * b31 + a34 * b41 = 0.00081 * 1 + -0.43281 * five + -0.10368 * 9 + 0.79488 * 13 = seven.23708; x32 = a31 * b12 + a32 * b22 + a33 * b32 + a34 * b42 = 0.00081 * 2 + -0.43281 * 6 + -0.10368 * 10 + 0.79488 * 14 = 7.49628; ten33 = a31 * b13 + a32 * b23 + a33 * b33 + a34 * b43 = 0.00081 * 3 + -0.43281 * 7 + -0.10368 * 11 + 0.79488 * fifteen = 7.75548; x34 = a31 * b14 + a32 * b24 + a33 * b34 + a34 * b44 = 0.00081 * four + -0.43281 * 8 + -0.10368 * 12 + 0.79488 * xvi = eight.01468; ten41 = a41 * b11 + a42 * b21 + a43 * b31 + a44 * b41 = 0.00567 * one + 0.26163 * 5 + 0.26163 * nine + -0.68931 * 13 = -5.29254; x42 = a41 * b12 + a42 * b22 + a43 * b32 + a44 * b42 = 0.00567 * 2 + 0.26163 * 6 + 0.26163 * 10 + -0.68931 * 14 = -5.45292; x43 = a41 * b13 + a42 * b23 + a43 * b33 + a44 * b43 = 0.00567 * three + 0.26163 * seven + 0.26163 * 11 + -0.68931 * fifteen = -5.6133; x44 = a41 * b14 + a42 * b24 + a43 * b34 + a44 * b44 = 0.00567 * 4 + 0.26163 * 8 + 0.26163 * 12 + -0.68931 * 16 = -5.77368;
Answer:
| 000 | -0.40095 | -0.4131 | -0.42525 | -0.4374 | 000 |
| -2.23209 | -2.21994 | -2.20779 | -2.19564 | ||
| 7.23708 | 7.49628 | 7.75548 | viii.01468 | ||
| -5.29254 | -5.45292 | -v.6133 | -5.77368 |
Solve The Matrix Equation Calculator,
Source: https://matrix-operations.com/main-matrix-equations
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