Matrix Operations: Determinant

Determinants

• Determinants are only applicable for square matrices.

• Determinant of the square matrix 𝐴 is denoted as:

det(𝐴) or 𝐴

• Recall that the absolute value of the determinant of a 2 × 2 matrix is equal to the area of parallelogram of the rows of that matrix.

• Similarly, the absolute value of the determinant of a 3 × 3 matrix is equal to the volume of parallelepiped of the rows of that matrix.

• Therefore, the absolute value of the determinant of a 𝑛 × 𝑛 matrix is equal to the n-dimensional volume, constructed by the rows of that matrix.

Determinant of a 2 × 2 matrix

• Recall that:

𝐴 =𝑎11 𝑎12𝑎21 𝑎22

, 𝐴 =𝑎11 𝑎12𝑎21 𝑎22

= 𝑎11𝑎22 − 𝑎12𝑎21.

𝒂1

𝒂2

𝒂𝟏𝒂𝟐

Determinant of a 3 × 3 matrix

• Also recall the determinant for a 3 ×3 matrix:

• 𝑅 =

𝑟11 𝑟12 𝑟13𝑟21 𝑟22 𝑟23𝑟31 𝑟32 𝑟33

• If the row vectors are linearlydependent, then the determinantis zero, and the matrix is NOT invertible.• Notice if the row vectors arelinearly dependent the volumewill be zero, as the vectors lie on a plane on a line.

𝒓1𝒓2𝒓3

Determinant of a 3 × 3 matrix• To compute the determinant of a 3 × 3 matrix,.

• The first element in the top row is multiplied with the determinant of the sub-matrix resulting from removing the (first) row and the (first) column corresponding to that element from the matrix.

• The negate of second element in the top row is multiplied with the determinant of the sub-matrix resulting from removing the (first) row and the (second) column corresponding to that element from the matrix.

• The third element in the top row is multiplied with the determinant of the sub-matrix resulting from removing the (first) row and the (third) column corresponding to that element from the matrix.

• 𝑅 =

𝑟11 𝑟12 𝑟13𝑟21 𝑟22 𝑟23𝑟31 𝑟32 𝑟33

= 𝑟11𝑟22 𝑟23𝑟32 𝑟33

− 𝑟12𝑟21 𝑟23𝑟31 𝑟33

+ 𝑟13𝑟21 𝑟22𝑟31 𝑟32

=

𝑟11 𝑟22𝑟33 − 𝑟23𝑟32 − 𝑟12 𝑟21𝑟33 − 𝑟23𝑟31 + 𝑟13 𝑟21𝑟32 − 𝑟22𝑟31

Determinant of a 3 × 3 matrix / Cofactor

• In the determinant of a 3 × 3 matrix, we multiplied the first row elements in their corresponding cofactors.

• The cofactor of the element 𝑖, 𝑗 of 𝑛 × 𝑛 matrix 𝐴 is:𝐶𝑖𝑗 = (−1)𝑖+𝑗det𝑀𝑖𝑗

• Where 𝑀𝑖𝑗 is submatrix after removing row 𝑖 and column 𝑗.• Determinant of 𝐴 is:

det𝐴 = 𝑎𝑖1𝐶𝑖1 + 𝑎𝑖2𝐶𝑖2 +⋯+ 𝑎𝑖𝑛𝐶𝑖𝑛• In the above formula the row 𝑖 could be any row of 𝐴 and it is not

necessarily the first row.• In fact it need not be a row. It can be any column j. • (So in order to compute the determinant, it is always wise to choose the

row or a column that has most number of zeroes and compute the cofactor of only its non-zero elements.)

Determinant properties

• The determinant of identity matrix is 1.𝐼 = 1

• The determinant changes sign when two rows are exchanged.𝑐 𝑑𝑎 𝑏

= −𝑎 𝑏𝑐 𝑑

• The determinant is a linear function of each row separately.𝑡𝑎 𝑡𝑏𝑐 𝑑

= 𝑡𝑎 𝑏𝑐 𝑑

𝑎 + 𝑎′ 𝑏 + 𝑏′

𝑐 𝑑=

𝑎 𝑏𝑐 𝑑

+𝑎′ 𝑏′

𝑐 𝑑

Determinant properties

• If one row is a scalar multiple of another row then det(𝐴) = 0

𝑎 𝑏𝑡𝑎 𝑡𝑏

= 0𝑎 𝑏 𝑐𝑑 𝑒 𝑓𝑡𝑎 𝑡𝑏 𝑡𝑐

= 0

𝑎 𝑏 𝑐𝑑 𝑒 𝑓

𝑎 + 𝑑 𝑏 + 𝑒 𝑐 + 𝑓= 0,

𝑎 𝑏 𝑐𝑑 𝑒 𝑓

2𝑎 + 𝑑 2𝑏 + 𝑒 2𝑐 + 𝑓= 0

𝑎 𝑏 𝑐𝑑 𝑒 𝑓

2𝑎 + 5𝑑 2𝑏 + 5𝑒 2𝑐 + 5𝑓= 0

Determinant properties

• Row reduction does not change the determinant of 𝐴𝑎 𝑏

𝑐 − 𝛾𝑎 𝑑 − 𝛾𝑏=

𝑎 𝑏𝑐 𝑑

𝛾 is a non-zero scalar

• A matrix with a row of zeros has det(𝐴) = 0𝑎 𝑏0 0

= 0

Determinant properties

• If 𝐴 is a triangular then the determinant is the product of diagonal elements.

𝑎 𝑏0 𝑑

= 𝑎𝑑,𝑎 0𝑐 𝑑

= 𝑎𝑑

This is also applicable for diagonal matrices:𝑎 0 00 𝑏 00 0 𝑐

= 𝑎𝑏𝑐

• If 𝐴 is singular (columns or rows are linearly dependent) det(𝐴) = 0

• 𝐴𝐵 = 𝐴 𝐵

• 𝐴𝑇 = 𝐴

Rank of Matrix

• Let 𝑚 = min 𝑟𝑜𝑤, 𝑐𝑜𝑙𝑢𝑚𝑛

• Rank of matrix is the size of the largest square sub-matrix with non-zero determinant.

• Matrix is full-ranked, if its rank = m.

• Matrix is rank-deficient, if its rank < m.

• It is not possible to have matrix’s rank > m.

Sub-Matrix

• In order to find the rank of matrix we should find the largest quaresub-matrix with non-zero determinant.

• For making a sub-matrix we are allowed to remove rows or columns of a matrix

• Example: A is a 5 × 3 matrix

• Removing two rows of A𝑟𝑜𝑤1𝑟𝑜𝑤2𝑟𝑜𝑤3𝑟𝑜𝑤4𝑟𝑜𝑤5

=𝑟𝑜𝑤2𝑟𝑜𝑤4𝑟𝑜𝑤5

Matrix Rank

• Example: Find the rank of matrix A

𝐴 =0 1 21 2 12 7 8

Row 1 and Row 2 of matrix A are linearly independent. However Row 3 is a linear combination of Row 1 and 2.

𝑟𝑜𝑤3 = 3 × 𝑟𝑜𝑤1 + 2 × 𝑟𝑜𝑤2

So A only have two independent row vectors. Now let remove third row and first column of A then we have a 2 × 2 matrix which determinant is not zero.

1 22 1

≠ 0

So rank of A is 2.