a 6x6 matrix with unkown x

i'm a civil engineering student and i'm having a problem solving a 6x6 matrix with an unkown x

the equation says ([K]-x[M])Φ=0

[M]=   [(841*x)/10,          0,          0,          0,          0,          0]
          [         0, (841*x)/10,          0,          0,          0,          0]
          [         0,          0, (841*x)/10,          0,          0,          0]
          [         0,          0,          0, (841*x)/10,          0,          0]
          [         0,          0,          0,          0, (841*x)/10,          0]
          [         0,          0,          0,          0,          0, (841*x)/10]
[K]=    673000    -455000    77600  -654000  455000   12600
          -455000   1810000   18000   445000  -321000   18000 
          77600       18000     429000  -12600   -18000     56875
         -654000   455000   -12600   1070000  -455000  507000
         455000   -321000   -18000   -455000   4480000 -18000
         12600       18000      56875   507000    -18000   997750

first of all i need to calculate det(K-M) to find the 6 values of x, then replace it in M to calculate the vectors Φ

if someone can help me or tell me the method, it is very urgent and it is for my senior project.

thank you guys.

Answer 1

[Matt J]

See eig().

[jad bousaid]

root(z1^6 + z1^5*((2523*x)/5 - 13320750) + z1^4*(- 5601375375*x + (2121843*x^2)/20 + 64922447204375) + z1^3*(21839911239551750*x - 942151338075*x^2 + (594823321*x^3)/50 - 142154701436104375000) + z1^2*(- 35865631172329133812500*x + (5510209605738906525*x^2)/2 - (158469855064215*x^3)/2 + (1500739238883*x^4)/2000 + 140014161262115304725000000) + z1*(23550381924287794254745000000*x - 3016299581592880153631250*x^2 + (308939085228428025835*x^3)/2 - (26654629621800963*x^4)/8 + (1262121699900603*x^5)/50000 - 52141801798740670429625000000000) + (353814783205469041*x^6)/1000000 + 990293559916301748412027250000*x^2 - (22416543511934609883*x^5)/400 - 4385125531274090383131462500000000*x - 84556931603987073640129375*x^3 + (51963554135421593945447*x^4)/16 + 2945824239263437512080650000000000000, z1, 1)

root(z1^6 + z1^5*((2523*x)/5 - 13320750) + z1^4*(- 5601375375*x + (2121843*x^2)/20 + 64922447204375) + z1^3*(21839911239551750*x - 942151338075*x^2 + (594823321*x^3)/50 - 142154701436104375000) + z1^2*(- 35865631172329133812500*x + (5510209605738906525*x^2)/2 - (158469855064215*x^3)/2 + (1500739238883*x^4)/2000 + 140014161262115304725000000) + z1*(23550381924287794254745000000*x - 3016299581592880153631250*x^2 + (308939085228428025835*x^3)/2 - (26654629621800963*x^4)/8 + (1262121699900603*x^5)/50000 - 52141801798740670429625000000000) + (353814783205469041*x^6)/1000000 + 990293559916301748412027250000*x^2 - (22416543511934609883*x^5)/400 - 4385125531274090383131462500000000*x - 84556931603987073640129375*x^3 + (51963554135421593945447*x^4)/16 + 2945824239263437512080650000000000000, z1, 2)

root(z1^6 + z1^5*((2523*x)/5 - 13320750) + z1^4*(- 5601375375*x + (2121843*x^2)/20 + 64922447204375) + z1^3*(21839911239551750*x - 942151338075*x^2 + (594823321*x^3)/50 - 142154701436104375000) + z1^2*(- 35865631172329133812500*x + (5510209605738906525*x^2)/2 - (158469855064215*x^3)/2 + (1500739238883*x^4)/2000 + 140014161262115304725000000) + z1*(23550381924287794254745000000*x - 3016299581592880153631250*x^2 + (308939085228428025835*x^3)/2 - (26654629621800963*x^4)/8 + (1262121699900603*x^5)/50000 - 52141801798740670429625000000000) + (353814783205469041*x^6)/1000000 + 990293559916301748412027250000*x^2 - (22416543511934609883*x^5)/400 - 4385125531274090383131462500000000*x - 84556931603987073640129375*x^3 + (51963554135421593945447*x^4)/16 + 2945824239263437512080650000000000000, z1, 3)

root(z1^6 + z1^5*((2523*x)/5 - 13320750) + z1^4*(- 5601375375*x + (2121843*x^2)/20 + 64922447204375) + z1^3*(21839911239551750*x - 942151338075*x^2 + (594823321*x^3)/50 - 142154701436104375000) + z1^2*(- 35865631172329133812500*x + (5510209605738906525*x^2)/2 - (158469855064215*x^3)/2 + (1500739238883*x^4)/2000 + 140014161262115304725000000) + z1*(23550381924287794254745000000*x - 3016299581592880153631250*x^2 + (308939085228428025835*x^3)/2 - (26654629621800963*x^4)/8 + (1262121699900603*x^5)/50000 - 52141801798740670429625000000000) + (353814783205469041*x^6)/1000000 + 990293559916301748412027250000*x^2 - (22416543511934609883*x^5)/400 - 4385125531274090383131462500000000*x - 84556931603987073640129375*x^3 + (51963554135421593945447*x^4)/16 + 2945824239263437512080650000000000000, z1, 4)

root(z1^6 + z1^5*((2523*x)/5 - 13320750) + z1^4*(- 5601375375*x + (2121843*x^2)/20 + 64922447204375) + z1^3*(21839911239551750*x - 942151338075*x^2 + (594823321*x^3)/50 - 142154701436104375000) + z1^2*(- 35865631172329133812500*x + (5510209605738906525*x^2)/2 - (158469855064215*x^3)/2 + (1500739238883*x^4)/2000 + 140014161262115304725000000) + z1*(23550381924287794254745000000*x - 3016299581592880153631250*x^2 + (308939085228428025835*x^3)/2 - (26654629621800963*x^4)/8 + (1262121699900603*x^5)/50000 - 52141801798740670429625000000000) + (353814783205469041*x^6)/1000000 + 990293559916301748412027250000*x^2 - (22416543511934609883*x^5)/400 - 4385125531274090383131462500000000*x - 84556931603987073640129375*x^3 + (51963554135421593945447*x^4)/16 + 2945824239263437512080650000000000000, z1, 5)

root(z1^6 + z1^5*((2523*x)/5 - 13320750) + z1^4*(- 5601375375*x + (2121843*x^2)/20 + 64922447204375) + z1^3*(21839911239551750*x - 942151338075*x^2 + (594823321*x^3)/50 - 142154701436104375000) + z1^2*(- 35865631172329133812500*x + (5510209605738906525*x^2)/2 - (158469855064215*x^3)/2 + (1500739238883*x^4)/2000 + 140014161262115304725000000) + z1*(23550381924287794254745000000*x - 3016299581592880153631250*x^2 + (308939085228428025835*x^3)/2 - (26654629621800963*x^4)/8 + (1262121699900603*x^5)/50000 - 52141801798740670429625000000000) + (353814783205469041*x^6)/1000000 + 990293559916301748412027250000*x^2 - (22416543511934609883*x^5)/400 - 4385125531274090383131462500000000*x - 84556931603987073640129375*x^3 + (51963554135421593945447*x^4)/16 + 2945824239263437512080650000000000000, z1, 6)

i can't find the roots, all i can get is that.