QuestionCalculate the determinant of the matrix: $\Delta=\left|\begin{array}{llll}1 & 2 & 3 & 4 \\ 3 & 5 & 7 & 9 \\ 4 & 7 & 11 & 15 \\ 8 & 4 & 21 & 30\end{array}\right|$

Studdy Solution

STEP 1

Assumptions1. We are asked to find the determinant of a4x4 matrix. . The matrix is given as follows $\Delta=\left|\begin{array}{llll}1 & &3 &4 \\3 &5 &7 &9 \\4 &7 &11 &15 \\8 &4 &21 &30\end{array}\right|$

STEP 2

We will use the method of minors and cofactors to find the determinant of this4x4 matrix. This involves finding the determinants ofx sub-matrices.

STEP 3

Let's start by expanding along the first row. The determinant of ax matrix can be calculated as follows $\Delta = a_{11}C_{11} - a_{12}C_{12} + a_{13}C_{13} - a_{14}C_{14}$ where $a_{ij}$ are the elements of the first row and $C_{ij}$ are the corresponding cofactors.

STEP 4

The cofactors $C_{ij}$ are calculated as the determinant of the (4-1)x(4-1) =3x3 matrix that results from removing the i-th row and j-th column from the original4x4 matrix, multiplied by $(-1)^{i+j}$ . Let's calculate these cofactors.

STEP 5

Calculate $C_{11}$ $C_{11} = (-1)^{1+1} \left|\begin{array}{lll}5 &7 &9 \\7 &11 &15 \\4 &21 &30\end{array}\right|$

STEP 6

Calculate $C_{12}$ $C_{12} = (-1)^{1+2} \left|\begin{array}{lll}3 & &9 \\4 &11 &15 \\8 &21 &30\end{array}\right|$

STEP 7

Calculate $C_{13}$ $C_{13} = (-1)^{1+3} \left|\begin{array}{lll}3 &5 &9 \\4 &7 &15 \\ &4 &30\end{array}\right|$

STEP 8

Calculate $C_{14}$ $C_{14} = (-1)^{1+4} \left|\begin{array}{lll}3 &5 &7 \\4 &7 &11 \\8 &4 &21\end{array}\right|$

STEP 9

Now we need to calculate the determinants of these3x3 matrices. We will use the formula for the determinant of a3x3 matrix $\left|\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right| = aei + bfg + cdh - ceg - bdi - afh$

STEP 10

Calculate the determinant for $C_{}$ $C_{} =5**30 +7154 +9721 -9**4 -71521 -5730$

STEP 11

Calculate the determinant for $C_{}$ $C_{} =31130 +7158 +9421 -9118 -7154 -3730$

STEP 12

Calculate the determinant for $C_{}$ $C_{} =730 +5158 +947 -978 -5154 -730$

STEP 13

Calculate the determinant for $C_{}$ $C_{} =3721 +5118 +7**7 -778 -511 -3721$

STEP 14

Now we can substitute the determinants of the cofactors back into the formula for the determinant of the4x4 matrix $\Delta =C_{11} -2C_{12} +3C_{13} -4C_{14}$

STEP 15

Plug in the values for $C_{11}$ , $C_{12}$ , $C_{13}$ , and $C_{14}$ to calculate the determinant.

STEP 16

Calculate the determinant of the matrix.
The final determinant value will be the solution to the problem.

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QuestionCalculate the determinant of the matrix: $\Delta=\left|\begin{array}{llll}1 & 2 & 3 & 4 \\ 3 & 5 & 7 & 9 \\ 4 & 7 & 11 & 15 \\ 8 & 4 & 21 & 30\end{array}\right|$

Studdy Solution

STEP 1

Assumptions1. We are asked to find the determinant of a4x4 matrix. . The matrix is given as follows $\Delta=\left|\begin{array}{llll}1 & &3 &4 \\3 &5 &7 &9 \\4 &7 &11 &15 \\8 &4 &21 &30\end{array}\right|$

STEP 2

We will use the method of minors and cofactors to find the determinant of this4x4 matrix. This involves finding the determinants ofx sub-matrices.

STEP 3

Let's start by expanding along the first row. The determinant of ax matrix can be calculated as follows $\Delta = a_{11}C_{11} - a_{12}C_{12} + a_{13}C_{13} - a_{14}C_{14}$ where $a_{ij}$ are the elements of the first row and $C_{ij}$ are the corresponding cofactors.

STEP 4

The cofactors $C_{ij}$ are calculated as the determinant of the (4-1)x(4-1) =3x3 matrix that results from removing the i-th row and j-th column from the original4x4 matrix, multiplied by $(-1)^{i+j}$ . Let's calculate these cofactors.

STEP 5

Calculate $C_{11}$ $C_{11} = (-1)^{1+1} \left|\begin{array}{lll}5 &7 &9 \\7 &11 &15 \\4 &21 &30\end{array}\right|$

STEP 6

Calculate $C_{12}$ $C_{12} = (-1)^{1+2} \left|\begin{array}{lll}3 & &9 \\4 &11 &15 \\8 &21 &30\end{array}\right|$

STEP 7

Calculate $C_{13}$ $C_{13} = (-1)^{1+3} \left|\begin{array}{lll}3 &5 &9 \\4 &7 &15 \\ &4 &30\end{array}\right|$

STEP 8

Calculate $C_{14}$ $C_{14} = (-1)^{1+4} \left|\begin{array}{lll}3 &5 &7 \\4 &7 &11 \\8 &4 &21\end{array}\right|$

STEP 9

Now we need to calculate the determinants of these3x3 matrices. We will use the formula for the determinant of a3x3 matrix $\left|\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right| = aei + bfg + cdh - ceg - bdi - afh$

STEP 10

Calculate the determinant for $C_{}$ $C_{} =5**30 +7154 +9721 -9**4 -71521 -5730$

STEP 11

Calculate the determinant for $C_{}$ $C_{} =31130 +7158 +9421 -9118 -7154 -3730$

STEP 12

Calculate the determinant for $C_{}$ $C_{} =730 +5158 +947 -978 -5154 -730$

STEP 13

Calculate the determinant for $C_{}$ $C_{} =3721 +5118 +7**7 -778 -511 -3721$

STEP 14

Now we can substitute the determinants of the cofactors back into the formula for the determinant of the4x4 matrix $\Delta =C_{11} -2C_{12} +3C_{13} -4C_{14}$

STEP 15

Plug in the values for $C_{11}$ , $C_{12}$ , $C_{13}$ , and $C_{14}$ to calculate the determinant.

STEP 16

Calculate the determinant of the matrix.
The final determinant value will be the solution to the problem.

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QuestionCalculate the determinant of the matrix: Δ=∣12343579471115842130∣\Delta=\left|\begin{array}{llll}1 & 2 & 3 & 4 \\ 3 & 5 & 7 & 9 \\ 4 & 7 & 11 & 15 \\ 8 & 4 & 21 & 30\end{array}\right|Δ=∣∣​1348​2574​371121​491530​∣∣​

Studdy Solution

STEP 1

Assumptions1. We are asked to find the determinant of a4x4 matrix. . The matrix is given as followsΔ=∣1343579471115842130∣\Delta=\left|\begin{array}{llll}1 & &3 &4 \\3 &5 &7 &9 \\4 &7 &11 &15 \\8 &4 &21 &30\end{array}\right|Δ=∣∣​1348​574​371121​491530​∣∣​

STEP 2

We will use the method of minors and cofactors to find the determinant of this4x4 matrix. This involves finding the determinants ofx sub-matrices.

STEP 3

STEP 4

The cofactors CijC_{ij}Cij​ are calculated as the determinant of the (4-1)x(4-1) =3x3 matrix that results from removing the i-th row and j-th column from the original4x4 matrix, multiplied by (−1)i+j(-1)^{i+j}(−1)i+j. Let's calculate these cofactors.

STEP 5

Calculate C11C_{11}C11​C11=(−1)1+1∣5797111542130∣C_{11} = (-1)^{1+1} \left|\begin{array}{lll}5 &7 &9 \\7 &11 &15 \\4 &21 &30\end{array}\right|C11​=(−1)1+1∣∣​574​71121​91530​∣∣​

STEP 6

Calculate C12C_{12}C12​C12=(−1)1+2∣394111582130∣C_{12} = (-1)^{1+2} \left|\begin{array}{lll}3 & &9 \\4 &11 &15 \\8 &21 &30\end{array}\right|C12​=(−1)1+2∣∣​348​1121​91530​∣∣​

STEP 7

Calculate C13C_{13}C13​C13=(−1)1+3∣3594715430∣C_{13} = (-1)^{1+3} \left|\begin{array}{lll}3 &5 &9 \\4 &7 &15 \\ &4 &30\end{array}\right|C13​=(−1)1+3∣∣​34​574​91530​∣∣​

STEP 8

Calculate C14C_{14}C14​C14=(−1)1+4∣35747118421∣C_{14} = (-1)^{1+4} \left|\begin{array}{lll}3 &5 &7 \\4 &7 &11 \\8 &4 &21\end{array}\right|C14​=(−1)1+4∣∣​348​574​71121​∣∣​

STEP 9

STEP 10

Calculate the determinant for CC_{}C​C=5∗∗30+7∗15∗4+9∗7∗21−9∗∗4−7∗15∗21−5∗7∗30C_{} =5**30 +7*15*4 +9*7*21 -9**4 -7*15*21 -5*7*30C​=5∗∗30+7∗15∗4+9∗7∗21−9∗∗4−7∗15∗21−5∗7∗30

STEP 11

Calculate the determinant for CC_{}C​C=3∗11∗30+7∗15∗8+9∗4∗21−9∗11∗8−7∗15∗4−3∗7∗30C_{} =3*11*30 +7*15*8 +9*4*21 -9*11*8 -7*15*4 -3*7*30C​=3∗11∗30+7∗15∗8+9∗4∗21−9∗11∗8−7∗15∗4−3∗7∗30

STEP 12

Calculate the determinant for CC_{}C​C=∗7∗30+5∗15∗8+9∗4∗7−9∗7∗8−5∗15∗4−∗7∗30C_{} =*7*30 +5*15*8 +9*4*7 -9*7*8 -5*15*4 -*7*30C​=∗7∗30+5∗15∗8+9∗4∗7−9∗7∗8−5∗15∗4−∗7∗30

STEP 13

Calculate the determinant for CC_{}C​C=3∗7∗21+5∗11∗8+7∗∗7−7∗7∗8−5∗11∗−3∗7∗21C_{} =3*7*21 +5*11*8 +7**7 -7*7*8 -5*11* -3*7*21C​=3∗7∗21+5∗11∗8+7∗∗7−7∗7∗8−5∗11∗−3∗7∗21

STEP 14

Now we can substitute the determinants of the cofactors back into the formula for the determinant of the4x4 matrixΔ=∗C11−2∗C12+3∗C13−4∗C14\Delta =*C_{11} -2*C_{12} +3*C_{13} -4*C_{14}Δ=∗C11​−2∗C12​+3∗C13​−4∗C14​

STEP 15

Plug in the values for C11C_{11}C11​, C12C_{12}C12​, C13C_{13}C13​, and C14C_{14}C14​ to calculate the determinant.

STEP 16

Calculate the determinant of the matrix.The final determinant value will be the solution to the problem.

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QuestionCalculate the determinant of the matrix: $\Delta=\left|\begin{array}{llll}1 & 2 & 3 & 4 \\ 3 & 5 & 7 & 9 \\ 4 & 7 & 11 & 15 \\ 8 & 4 & 21 & 30\end{array}\right|$

Assumptions1. We are asked to find the determinant of a4x4 matrix. . The matrix is given as follows $\Delta=\left|\begin{array}{llll}1 & &3 &4 \\3 &5 &7 &9 \\4 &7 &11 &15 \\8 &4 &21 &30\end{array}\right|$

The cofactors $C_{ij}$ are calculated as the determinant of the (4-1)x(4-1) =3x3 matrix that results from removing the i-th row and j-th column from the original4x4 matrix, multiplied by $(-1)^{i+j}$ . Let's calculate these cofactors.

Calculate $C_{11}$ $C_{11} = (-1)^{1+1} \left|\begin{array}{lll}5 &7 &9 \\7 &11 &15 \\4 &21 &30\end{array}\right|$

Calculate $C_{12}$ $C_{12} = (-1)^{1+2} \left|\begin{array}{lll}3 & &9 \\4 &11 &15 \\8 &21 &30\end{array}\right|$

Calculate $C_{13}$ $C_{13} = (-1)^{1+3} \left|\begin{array}{lll}3 &5 &9 \\4 &7 &15 \\ &4 &30\end{array}\right|$

Calculate $C_{14}$ $C_{14} = (-1)^{1+4} \left|\begin{array}{lll}3 &5 &7 \\4 &7 &11 \\8 &4 &21\end{array}\right|$

Calculate the determinant for $C_{}$ $C_{} =5**30 +7154 +9721 -9**4 -71521 -5730$

Calculate the determinant for $C_{}$ $C_{} =31130 +7158 +9421 -9118 -7154 -3730$

Calculate the determinant for $C_{}$ $C_{} =730 +5158 +947 -978 -5154 -730$

Calculate the determinant for $C_{}$ $C_{} =3721 +5118 +7**7 -778 -511 -3721$

Now we can substitute the determinants of the cofactors back into the formula for the determinant of the4x4 matrix $\Delta =C_{11} -2C_{12} +3C_{13} -4C_{14}$

Plug in the values for $C_{11}$ , $C_{12}$ , $C_{13}$ , and $C_{14}$ to calculate the determinant.

Calculate the determinant of the matrix.
The final determinant value will be the solution to the problem.