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Question: Expand: \[\left| \begin{matrix} 1 & -7 & 3 \\\ 5 & -6 & 0 \\\ 1 & 2 & -3 \\\ \end{m...

Expand: ∣1−73 5−60 12−3 ∣\left| \begin{matrix} 1 & -7 & 3 \\\ 5 & -6 & 0 \\\ 1 & 2 & -3 \\\ \end{matrix} \right|

Explanation

Solution

Hint: To find the value of a given matrix, use the formula for expansion of value of the matrix and substitute the values to get the value of expansion.

We have the matrix∣1−73 5−60 12−3 ∣\left| \begin{matrix} 1 & -7 & 3 \\\ 5 & -6 & 0 \\\ 1 & 2 & -3 \\\ \end{matrix} \right|. We observe that this is a 3×33\times 3matrix.
Matrix is a rectangular array of numbers, symbols or expressions, arranged in rows and columns. Provided that two matrices have the same size (each matrix has the same number of rows and same number of columns as the other), two matrices can be added or subtracted element by element. The rule for matrix multiplication is that two matrices can be multiplied only when the number of columns in the first matrix equals the number of rows in the second one. Also, matrix multiplication is not commutative.
Any matrix is represented as A=[aij]m×nA={{\left[ {{a}_{ij}} \right]}_{m\times n}} which has mm rows and nn columns.
We know that the formula for expansion of the matrix ∣abc def ghi ∣\left| \begin{matrix} a & b & c \\\ d & e & f \\\ g & h & i \\\ \end{matrix} \right|isa(ei−hf)−b(di−fg)+c(dh−ge)a\left( ei-hf \right)-b\left( di-fg \right)+c\left( dh-ge \right).
Substituting the value a=1,b=−7,c=3,d=5,e=−6,f=0,g=1,h=2,i=−3a=1,b=-7,c=3,d=5,e=-6,f=0,g=1,h=2,i=-3, we have ∣1−73 5−60 12−3 ∣=1[(−6)(−3)−(0)(2)]−(−7)[(5)(−3)−(0)(1)]+3[(5)(2)−(1)(−6)]\left| \begin{matrix} 1 & -7 & 3 \\\ 5 & -6 & 0 \\\ 1 & 2 & -3 \\\ \end{matrix} \right|=1\left[ \left( -6 \right)\left( -3 \right)-\left( 0 \right)\left( 2 \right) \right]-\left( -7 \right)\left[ \left( 5 \right)\left( -3 \right)-\left( 0 \right)\left( 1 \right) \right]+3\left[ \left( 5 \right)\left( 2 \right)-\left( 1 \right)\left( -6 \right) \right].
Simplifying the above equation, we have∣1−73 5−60 12−3 ∣=1(18−0)+7(−15−0)+3(10+6)=18−105+48=−39\left| \begin{matrix} 1 & -7 & 3 \\\ 5 & -6 & 0 \\\ 1 & 2 & -3 \\\ \end{matrix} \right|=1\left( 18-0 \right)+7\left( -15-0 \right)+3\left( 10+6 \right)=18-105+48=-39.
Thus, we have the value of the given matrix as−39-39.
This method of matrix expansion of matrix is called cofactor expansion. It is an expression for the weighted sum of the determinants of the sub matrices of the given matrix. For a matrix of the form A=[aij]n×nA={{\left[ {{a}_{ij}} \right]}_{n\times n}}, we define Cij=(−1)i+jMij{{C}_{ij}}={{\left( -1 \right)}^{i+j}}{{M}_{ij}} as the cofactor of matrix AA where Mij{{M}_{ij}} is the i,ji,j minor of AA .
We define the determinant of AA as ∣A∣=ai1Ci1+ai2Ci2+...+ainCin=a1iC1i+a2iC2i+...aniCni\left| A \right|={{a}_{i1}}{{C}_{i1}}+{{a}_{i2}}{{C}_{i2}}+...+{{a}_{in}}{{C}_{in}}={{a}_{1i}}{{C}_{1i}}+{{a}_{2i}}{{C}_{2i}}+...{{a}_{ni}}{{C}_{ni}}. Thus, we observe that we can expand a matrix in multiple ways along its various rows and columns.

Note: We must keep in mind that we can expand any matrix along various rows and columns. But we will get the same value of the matrix by expanding along any row or column.