الجبر الخطي الأمثلة

أوجد معكوس المصفوفة الناتجة [[8,93,0.01],[44,575,5],[1,0.1,0.1]][[8,24,5],[8,9,6],[0.1,1,2]]
[8930.0144575510.10.1][82458960.112]8930.0144575510.10.182458960.112
خطوة 1
اضرب [8930.0144575510.10.1][82458960.112].
انقر لعرض المزيد من الخطوات...
خطوة 1.1
Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is 3×3 and the second matrix is 3×3.
خطوة 1.2
اضرب كل صف في المصفوفة الأولى في كل عمود في المصفوفة الثانية.
[88+938+0.010.1824+939+0.01185+936+0.012448+5758+50.14424+5759+51445+5756+5218+0.18+0.10.1124+0.19+0.1115+0.16+0.12]
خطوة 1.3
بسّط كل عنصر من عناصر المصفوفة بضرب جميع العبارات.
[808.0011029.01598.024952.5623636808.81255.8]
[808.0011029.01598.024952.5623636808.81255.8]
خطوة 2
Find the determinant.
انقر لعرض المزيد من الخطوات...
خطوة 2.1
Choose the row or column with the most 0 elements. If there are no 0 elements choose any row or column. Multiply every element in row 1 by its cofactor and add.
انقر لعرض المزيد من الخطوات...
خطوة 2.1.1
Consider the corresponding sign chart.
|+-+-+-+-+|
خطوة 2.1.2
The cofactor is the minor with the sign changed if the indices match a - position on the sign chart.
خطوة 2.1.3
The minor for a11 is the determinant with row 1 and column 1 deleted.
|62363680255.8|
خطوة 2.1.4
Multiply element a11 by its cofactor.
808.001|62363680255.8|
خطوة 2.1.5
The minor for a12 is the determinant with row 1 and column 2 deleted.
|4952.536808.815.8|
خطوة 2.1.6
Multiply element a12 by its cofactor.
-1029.01|4952.536808.815.8|
خطوة 2.1.7
The minor for a13 is the determinant with row 1 and column 3 deleted.
|4952.562368.8125|
خطوة 2.1.8
Multiply element a13 by its cofactor.
598.02|4952.562368.8125|
خطوة 2.1.9
Add the terms together.
808.001|62363680255.8|-1029.01|4952.536808.815.8|+598.02|4952.562368.8125|
808.001|62363680255.8|-1029.01|4952.536808.815.8|+598.02|4952.562368.8125|
خطوة 2.2
احسِب قيمة |62363680255.8|.
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خطوة 2.2.1
يمكن إيجاد محدد المصفوفة 2×2 باستخدام القاعدة |abcd|=ad-cb.
808.001(62365.8-253680)-1029.01|4952.536808.815.8|+598.02|4952.562368.8125|
خطوة 2.2.2
بسّط المحدد.
انقر لعرض المزيد من الخطوات...
خطوة 2.2.2.1
بسّط كل حد.
انقر لعرض المزيد من الخطوات...
خطوة 2.2.2.1.1
اضرب 6236 في 5.8.
808.001(36168.8-253680)-1029.01|4952.536808.815.8|+598.02|4952.562368.8125|
خطوة 2.2.2.1.2
اضرب -25 في 3680.
808.001(36168.8-92000)-1029.01|4952.536808.815.8|+598.02|4952.562368.8125|
808.001(36168.8-92000)-1029.01|4952.536808.815.8|+598.02|4952.562368.8125|
خطوة 2.2.2.2
اطرح 92000 من 36168.8.
808.001-55831.2-1029.01|4952.536808.815.8|+598.02|4952.562368.8125|
808.001-55831.2-1029.01|4952.536808.815.8|+598.02|4952.562368.8125|
808.001-55831.2-1029.01|4952.536808.815.8|+598.02|4952.562368.8125|
خطوة 2.3
احسِب قيمة |4952.536808.815.8|.
انقر لعرض المزيد من الخطوات...
خطوة 2.3.1
يمكن إيجاد محدد المصفوفة 2×2 باستخدام القاعدة |abcd|=ad-cb.
808.001-55831.2-1029.01(4952.55.8-8.813680)+598.02|4952.562368.8125|
خطوة 2.3.2
بسّط المحدد.
انقر لعرض المزيد من الخطوات...
خطوة 2.3.2.1
بسّط كل حد.
انقر لعرض المزيد من الخطوات...
خطوة 2.3.2.1.1
اضرب 4952.5 في 5.8.
808.001-55831.2-1029.01(28724.5-8.813680)+598.02|4952.562368.8125|
خطوة 2.3.2.1.2
اضرب -8.81 في 3680.
808.001-55831.2-1029.01(28724.5-32420.8)+598.02|4952.562368.8125|
808.001-55831.2-1029.01(28724.5-32420.8)+598.02|4952.562368.8125|
خطوة 2.3.2.2
اطرح 32420.8 من 28724.5.
808.001-55831.2-1029.01-3696.3+598.02|4952.562368.8125|
808.001-55831.2-1029.01-3696.3+598.02|4952.562368.8125|
808.001-55831.2-1029.01-3696.3+598.02|4952.562368.8125|
خطوة 2.4
احسِب قيمة |4952.562368.8125|.
انقر لعرض المزيد من الخطوات...
خطوة 2.4.1
يمكن إيجاد محدد المصفوفة 2×2 باستخدام القاعدة |abcd|=ad-cb.
808.001-55831.2-1029.01-3696.3+598.02(4952.525-8.816236)
خطوة 2.4.2
بسّط المحدد.
انقر لعرض المزيد من الخطوات...
خطوة 2.4.2.1
بسّط كل حد.
انقر لعرض المزيد من الخطوات...
خطوة 2.4.2.1.1
اضرب 4952.5 في 25.
808.001-55831.2-1029.01-3696.3+598.02(123812.5-8.816236)
خطوة 2.4.2.1.2
اضرب -8.81 في 6236.
808.001-55831.2-1029.01-3696.3+598.02(123812.5-54939.16)
808.001-55831.2-1029.01-3696.3+598.02(123812.5-54939.16)
خطوة 2.4.2.2
اطرح 54939.16 من 123812.5.
808.001-55831.2-1029.01-3696.3+598.0268873.34
808.001-55831.2-1029.01-3696.3+598.0268873.34
808.001-55831.2-1029.01-3696.3+598.0268873.34
خطوة 2.5
بسّط المحدد.
انقر لعرض المزيد من الخطوات...
خطوة 2.5.1
بسّط كل حد.
انقر لعرض المزيد من الخطوات...
خطوة 2.5.1.1
اضرب 808.001 في -55831.2.
-45111665.4312-1029.01-3696.3+598.0268873.34
خطوة 2.5.1.2
اضرب -1029.01 في -3696.3.
-45111665.4312+3803529.663+598.0268873.34
خطوة 2.5.1.3
اضرب 598.02 في 68873.34.
-45111665.4312+3803529.663+41187634.7868
-45111665.4312+3803529.663+41187634.7868
خطوة 2.5.2
أضف -45111665.4312 و3803529.663.
-41308135.7682+41187634.7868
خطوة 2.5.3
أضف -41308135.7682 و41187634.7868.
-120500.9814
-120500.9814
-120500.9814
خطوة 3
Since the determinant is non-zero, the inverse exists.
خطوة 4
Set up a 3×6 matrix where the left half is the original matrix and the right half is its identity matrix.
[808.0011029.01598.021004952.5623636800108.81255.8001]
خطوة 5
أوجِد الصيغة الدرجية المختزلة صفيًا.
انقر لعرض المزيد من الخطوات...
خطوة 5.1
Multiply each element of R1 by 1808.001 to make the entry at 1,1 a 1.
انقر لعرض المزيد من الخطوات...
خطوة 5.1.1
Multiply each element of R1 by 1808.001 to make the entry at 1,1 a 1.
[808.001808.0011029.01808.001598.02808.0011808.0010808.0010808.0014952.5623636800108.81255.8001]
خطوة 5.1.2
بسّط R1.
[11.273525650.740122840.00123762004952.5623636800108.81255.8001]
[11.273525650.740122840.00123762004952.5623636800108.81255.8001]
خطوة 5.2
Perform the row operation R2=R2-4952.5R1 to make the entry at 2,1 a 0.
انقر لعرض المزيد من الخطوات...
خطوة 5.2.1
Perform the row operation R2=R2-4952.5R1 to make the entry at 2,1 a 0.
[11.273525650.740122840.00123762004952.5-4952.516236-4952.51.273525653680-4952.50.740122840-4952.50.001237621-4952.500-4952.508.81255.8001]
خطوة 5.2.2
بسّط R2.
[11.273525650.740122840.00123762000-71.1357894314.54160329-6.12932409108.81255.8001]
[11.273525650.740122840.00123762000-71.1357894314.54160329-6.12932409108.81255.8001]
خطوة 5.3
Perform the row operation R3=R3-8.81R1 to make the entry at 3,1 a 0.
انقر لعرض المزيد من الخطوات...
خطوة 5.3.1
Perform the row operation R3=R3-8.81R1 to make the entry at 3,1 a 0.
[11.273525650.740122840.00123762000-71.1357894314.54160329-6.12932409108.81-8.81125-8.811.273525655.8-8.810.740122840-8.810.001237620-8.8101-8.810]
خطوة 5.3.2
بسّط R3.
[11.273525650.740122840.00123762000-71.1357894314.54160329-6.1293240910013.780239-0.72048227-0.0109034501]
[11.273525650.740122840.00123762000-71.1357894314.54160329-6.1293240910013.780239-0.72048227-0.0109034501]
خطوة 5.4
Multiply each element of R2 by 1-71.13578943 to make the entry at 2,2 a 1.
انقر لعرض المزيد من الخطوات...
خطوة 5.4.1
Multiply each element of R2 by 1-71.13578943 to make the entry at 2,2 a 1.
[11.273525650.740122840.00123762000-71.13578943-71.13578943-71.1357894314.54160329-71.13578943-6.12932409-71.135789431-71.135789430-71.13578943013.780239-0.72048227-0.0109034501]
خطوة 5.4.2
بسّط R2.
[11.273525650.740122840.001237620001-0.204420350.08616371-0.014057620013.780239-0.72048227-0.0109034501]
[11.273525650.740122840.001237620001-0.204420350.08616371-0.014057620013.780239-0.72048227-0.0109034501]
خطوة 5.5
Perform the row operation R3=R3-13.780239R2 to make the entry at 3,2 a 0.
انقر لعرض المزيد من الخطوات...
خطوة 5.5.1
Perform the row operation R3=R3-13.780239R2 to make the entry at 3,2 a 0.
[11.273525650.740122840.001237620001-0.204420350.08616371-0.0140576200-13.780239013.780239-13.7802391-0.72048227-13.780239-0.20442035-0.01090345-13.7802390.086163710-13.780239-0.014057621-13.7802390]
خطوة 5.5.2
بسّط R3.
[11.273525650.740122840.001237620001-0.204420350.08616371-0.014057620002.09647906-1.198260070.193717381]
[11.273525650.740122840.001237620001-0.204420350.08616371-0.014057620002.09647906-1.198260070.193717381]
خطوة 5.6
Multiply each element of R3 by 12.09647906 to make the entry at 3,3 a 1.
انقر لعرض المزيد من الخطوات...
خطوة 5.6.1
Multiply each element of R3 by 12.09647906 to make the entry at 3,3 a 1.
[11.273525650.740122840.001237620001-0.204420350.08616371-0.01405762002.0964790602.096479062.096479062.09647906-1.198260072.096479060.193717382.0964790612.09647906]
خطوة 5.6.2
بسّط R3.
[11.273525650.740122840.001237620001-0.204420350.08616371-0.014057620001-0.571558330.092401290.47699021]
[11.273525650.740122840.001237620001-0.204420350.08616371-0.014057620001-0.571558330.092401290.47699021]
خطوة 5.7
Perform the row operation R2=R2+0.20442035R3 to make the entry at 2,3 a 0.
انقر لعرض المزيد من الخطوات...
خطوة 5.7.1
Perform the row operation R2=R2+0.20442035R3 to make the entry at 2,3 a 0.
[11.273525650.740122840.00123762000+0.2044203501+0.204420350-0.20442035+0.2044203510.08616371+0.20442035-0.57155833-0.01405762+0.204420350.092401290+0.204420350.47699021001-0.571558330.092401290.47699021]
خطوة 5.7.2
بسّط R2.
[11.273525650.740122840.0012376200010-0.030674430.004831080.0975065001-0.571558330.092401290.47699021]
[11.273525650.740122840.0012376200010-0.030674430.004831080.0975065001-0.571558330.092401290.47699021]
خطوة 5.8
Perform the row operation R1=R1-0.74012284R3 to make the entry at 1,3 a 0.
انقر لعرض المزيد من الخطوات...
خطوة 5.8.1
Perform the row operation R1=R1-0.74012284R3 to make the entry at 1,3 a 0.
[1-0.7401228401.27352565-0.7401228400.74012284-0.7401228410.00123762-0.74012284-0.571558330-0.740122840.092401290-0.740122840.47699021010-0.030674430.004831080.0975065001-0.571558330.092401290.47699021]
خطوة 5.8.2
بسّط R1.
[11.2735256500.424261-0.06838831-0.35303135010-0.030674430.004831080.0975065001-0.571558330.092401290.47699021]
[11.2735256500.424261-0.06838831-0.35303135010-0.030674430.004831080.0975065001-0.571558330.092401290.47699021]
خطوة 5.9
Perform the row operation R1=R1-1.27352565R2 to make the entry at 1,2 a 0.
انقر لعرض المزيد من الخطوات...
خطوة 5.9.1
Perform the row operation R1=R1-1.27352565R2 to make the entry at 1,2 a 0.
[1-1.2735256501.27352565-1.2735256510-1.2735256500.424261-1.27352565-0.03067443-0.06838831-1.273525650.00483108-0.35303135-1.273525650.0975065010-0.030674430.004831080.0975065001-0.571558330.092401290.47699021]
خطوة 5.9.2
بسّط R1.
[1000.46332568-0.07454082-0.47720839010-0.030674430.004831080.0975065001-0.571558330.092401290.47699021]
[1000.46332568-0.07454082-0.47720839010-0.030674430.004831080.0975065001-0.571558330.092401290.47699021]
[1000.46332568-0.07454082-0.47720839010-0.030674430.004831080.0975065001-0.571558330.092401290.47699021]
خطوة 6
The right half of the reduced row echelon form is the inverse.
[0.46332568-0.07454082-0.47720839-0.030674430.004831080.0975065-0.571558330.092401290.47699021]
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