Lecture 1

Review of Matrix Algebra

 Terminology
Matrix

• Definition: A matrix is a rectangular array of numbers arranged into rows

and columns. It is written as
𝑎11 ⋯ 𝑎1𝑛
⋮ ⋱ ⋮
𝑎𝑚1 ⋯ 𝑎𝑚𝑛
• The above array is called an m by n (m×n) matrix.
• The number in row i and column j, 𝑎𝑖𝑗 , is called the (i,j)th entry.
• The matrix A is often denoted as
𝐴 = 𝑎𝑖𝑗
• The size of a matrix is indicated by the number of its rows and the number
of columns. A matrix with m rows and n columns is called an 𝑚 × 𝑛 matrix.
 Terminology
Vector

• Definition: A vector is a n-tuple of numbers. In ℛ3 a vector is a 3-tuple, i.e.,

(1,2,3). Similarly for ℛ𝑛 .Vectors are usually denoted by lower case letters
such as a or b.
• Row Vector: A matrix with one row and 𝑛 columns (1 × 𝑛) is called a row
vector. It is usually written as
𝒙 = 𝑥1 , 𝑥2 , … , 𝑥𝑛
• Column Vector: A matrix with 𝑛 rows and 1 column (𝑛 × 1) is called a
column vector. It is usually written as
𝑥1
𝑥2
𝒙= ⋮
𝑥𝑛
 Algebraic Manipulation of Matrices

• Equality of matrices: Two matrices are equal if they both have the same
size and if the corresponding entries in the two matrices are equal. For
example,
1 2 3 1 2 1 2 1 1
≠ , ≠
2 3 4 2 3 2 3 2 3
• Addition: One can add two matrices of the same size, which is to say, with
the same number of rows and columns. Their sum is a new matrix with the
same size with the two matrices being added. The (i,j)th entry of the sum
matrix is the sum of the (i,j)th entry of the two matrices being added. In
symbols
𝑎11 ⋯ 𝑎1𝑛 𝑏11 ⋯ 𝑏1𝑛
⋮ 𝑎𝑖𝑗 ⋮ + ⋮ 𝑏𝑖𝑗 ⋮
𝑎𝑚1 ⋯ 𝑎𝑚𝑛 𝑏𝑚1 ⋯ 𝑏𝑚𝑛
𝑎11 + 𝑏11 ⋯ 𝑎1𝑛 + 𝑏1𝑛
= ⋮ 𝑎𝑖𝑗 + 𝑏𝑖𝑗 ⋮
𝑎𝑚1 + 𝑏𝑚1 ⋯ 𝑎𝑚𝑛 + 𝑏𝑚𝑛
 Algebraic Manipulation of Matrices

• −𝐴 is defined as 𝐴 + −𝐴 = 0. That is
𝑎11 ⋯ 𝑎1𝑛 −𝑎11 ⋯ −𝑎1𝑛
− ⋮ 𝑎𝑖𝑗 ⋮ = ⋮ −𝑎𝑖𝑗 ⋮
𝑎𝑚1 ⋯ 𝑎𝑚𝑛 −𝑎𝑚1 ⋯ −𝑎𝑚𝑛
• Subtraction: 𝐴 − 𝐵 = 𝐴 + −𝐵 , we subtract matrices of the same size
simply by subtracting their corresponding entries.
𝑎11 ⋯ 𝑎1𝑛 𝑏11 ⋯ 𝑏1𝑛
⋮ 𝑎𝑖𝑗 ⋮ − ⋮ 𝑏𝑖𝑗 ⋮
𝑎𝑚1 ⋯ 𝑎𝑚𝑛 𝑏𝑚1 ⋯ 𝑏𝑚𝑛
𝑎11 − 𝑏11 ⋯ 𝑎1𝑛 − 𝑏1𝑛
= ⋮ 𝑎𝑖𝑗 − 𝑏𝑖𝑗 ⋮
𝑎𝑚1 − 𝑏𝑚1 ⋯ 𝑎𝑚𝑛 − 𝑏𝑚𝑛
 Algebraic Manipulation of Matrices

• Scalar multiplication: The product of a matrix 𝐴 and a number 𝑟, denoted

by 𝑟𝐴, is a matrix created by multiplying each entry of 𝐴 by 𝑟.
𝑎11 ⋯ 𝑎1𝑛 𝑟𝑎11 ⋯ 𝑟𝑎1𝑛
𝑟 ⋮ 𝑎𝑖𝑗 ⋮ = ⋮ 𝑟𝑎𝑖𝑗 ⋮
𝑎𝑚1 ⋯ 𝑎𝑚𝑛 𝑟𝑎𝑚1 ⋯ 𝑟𝑎𝑚𝑛
• In summary, within the class of 𝑚 × 𝑛 matrices, addition, subtraction and
scalar multiplication are all defined in terms of the corresponding entry of
the matrices.
• Matrix multiplication: We define the matrix product 𝐴𝐵 if and only if
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 𝑜𝑓 𝐴 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑜𝑤𝑠 𝑜𝑓 𝐵
The (𝑖, 𝑗)𝑡ℎ entry of 𝐴𝐵, multiply the 𝑖𝑡ℎ row of 𝐴 and 𝑗𝑡ℎ column of 𝐵 as
follows:
𝑏1𝑗
𝑛
𝑏2𝑗
𝑎𝑖1 , 𝑎𝑖2 , … , 𝑎𝑖𝑛 = 𝑎𝑖1 𝑏1𝑗 + 𝑎𝑖2 𝑏2𝑗 + ⋯ + 𝑎𝑖𝑛 𝑏𝑛𝑗 = ෍ 𝑎𝑖ℎ 𝑏ℎ𝑗
⋮
ℎ=1
𝑏𝑛𝑗
 Algebraic Manipulation of Matrices

• For Example
𝑎 𝑏 𝑎𝐴 + 𝑏𝐶 𝑎𝐵 + 𝑏𝐷
𝑐 𝑑 𝐴 𝐵
= 𝑐𝐴 + 𝑑𝐶 𝑐𝐵 + 𝑑𝐷
𝑒 𝑓 𝐶 𝐷 𝑒𝐴 + 𝑓𝐶 𝑒𝐵 + 𝑓𝐷
Note that is this case, the product taken in reverse order,
𝑎 𝑏
𝐴 𝐵 𝑐 𝑑
𝐶 𝐷 𝑒 𝑓

is not defined.
• If A is 𝑚 × 𝑛, and 𝐵 is 𝑛 × 𝑘, then the product 𝐴𝐵 is 𝑚 × 𝑘. The product
matrix 𝐴𝐵 inherits the number of its rows from 𝐴 and the number of its
columns from 𝐵.

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑜𝑤𝑠 𝑜𝑓 𝐴𝐵 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑜𝑤𝑠 𝑜𝑓 𝐴

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 𝑜𝑓 𝐴𝐵 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 𝑜𝑓 𝐵
𝑚×𝑛 ⋅ 𝑛×𝑘 = 𝑚×𝑘
 Algebraic Manipulation of Matrices

• Transpose: The transpose of a 𝑚 × 𝑛 matrix 𝐴 is a 𝑛 × 𝑚 matrix obtained

by interchanging the rows and columns of 𝐴.
– This matrix is often written as 𝐴𝑇 or 𝐴′.
– The first row of 𝐴 becomes the first column of 𝐴𝑇 , the second row of 𝐴
becomes the second column of 𝐴𝑇 , and so on. Thus the 𝑖, 𝑗 𝑡ℎ entry of
𝐴 becomes the 𝑗, 𝑖 𝑡ℎ entry of 𝐴𝑇 .
– For example
𝑎11 𝑎21
𝑎11 𝑎12 𝑎13 𝑇
𝑎 𝑎22
𝑎21 𝑎22 𝑎23 = 12
𝑎13 𝑎23
 Laws of Matrix Algebra

• Associative Laws:
𝐴+𝐵 +𝐶 =𝐴+ 𝐵+𝐶
𝐴𝐵 𝐶 = 𝐴 𝐵𝐶
• Commutative Law for Addition:
𝐴+𝐵 =𝐵+𝐴
• Distributive Laws:
𝐴 𝐵 + 𝐶 = 𝐴𝐵 + 𝐴𝐶
𝐴 + 𝐵 𝐶 = 𝐴𝐶 + 𝐵𝐶
• Transpose of a product:
𝐴𝐵 𝑇 = 𝐵𝑇 𝐴𝑇
• Transpose of a extended product:
𝐴𝐵𝐶 𝑇 = 𝐶 𝑇 𝐵𝑇 𝐴𝑇
 Systems of Equations in Matrix Form

• Consider the system of linear equations

𝑎11 𝑥1 + ⋯ + 𝑎1𝑛 𝑥𝑛 = 𝑏1
𝑎21 𝑥1 + ⋯ + 𝑎2𝑛 𝑥𝑛 = 𝑏2
⋮
𝑎𝑚1 𝑥1 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 = 𝑏𝑚
• This system can be expressed much more compactly using the notation
suggested by matrix algebra.
𝐴𝒙 = 𝑏
where
𝑎11 ⋯ 𝑎1𝑛
𝐴= ⋮ 𝑎𝑖𝑗 ⋮
𝑎𝑚1 ⋯ 𝑎𝑚𝑛
 Systems of Equations in Matrix Form

𝑥1 𝑏1
𝒙= ⋮ 𝑎𝑛𝑑 𝒃 = ⋮
𝑥𝑛 𝑏𝑚
𝐴 contains all of the coefficients, 𝒙 contains all the unknown variables, 𝒃
contains the constant terms in the right hand side of the equation system.
 Special Matrices

• Square Matrix: 𝑚 = 𝑛, that is equal number of rows and columns.

• Column Matrix: n=1, that is one column. For example
𝑎
𝑏 and 0 .
𝑐 1
• Row Matrix: 𝑚 = 1,that is one row. For example
2 1 0 and 2 3
• Diagonal Matrix: 𝑚 = 𝑛 and 𝑎𝑖𝑗 = 0 for 𝑖 ≠ 𝑗, that is a square matrix in
which all nondiagonal entries are 0. For example
1 0 0
𝑎 0
and 0 2 0
0 𝑏
0 0 3
 Special Matrices

• Identity Matrix: The 𝑛 × 𝑛 matrix

1 0 ⋯ 0
𝐼= 0 1 ⋯ 0
⋮ ⋮ ⋱ ⋮
0 0 ⋯ 1
with 𝑎𝑖𝑖 = 1for all 𝑖 and 𝑎𝑖𝑗 = 0 for all 𝑖 ≠ 𝑗.
• For any 𝑚 × 𝑛 matrix 𝐴,
𝐴𝐼 = 𝐴
• For any 𝑛 × 𝑘 matrix 𝐵,
𝐼𝐵 = 𝐵
• The matrix 𝐼 is called the 𝑛 × 𝑛 identity.
 Special Matrix

• Upper-Triangular Matrix: 𝑎𝑖𝑗 = 0 if 𝑖 > 𝑗, that is a matrix (usually

square) in which all entries below the diagonal are 0. For example
1 2 3
𝑎 𝑏
and 0 4 5
0 𝑑
0 0 6
• Lower-Triangular Matrix: 𝑎𝑖𝑗 = 0 if 𝑖 < 𝑗, that is a matrix (usually
square) in which all entries above the diagonal are 0. For example
1 0 0
𝑎 0
and 2 3 0
𝑐 𝑑
4 5 6
 Special Matrices

• Symmetric Matrix: 𝐴𝑇 = 𝐴, that is 𝑎𝑖𝑗 = 𝑎𝑗𝑖 , for all 𝑖, 𝑗. These matrices are
necessarily square. For example
1 2 3
𝑎 𝑏
and 2 4 5
𝑏 𝑑
3 5 6
• Idempotent Matrix: A square matrix 𝐵 for which 𝐵 ∙ 𝐵=B, such as 𝐵 = 𝐼 or
5 −5
4 −4
• Permutation Matrix: A square matrix of 0s and 1s in which each row and
each column contains exactly one 1. For example
0 1 0
1 0 0
0 0 1
 Elementary Matrices

• Recall three elementary row operations:

– Interchanging rows
– Adding a multiple of one row to another
– Multiplying a row by a nonzero scalar
• These operations can be performed on a matrix 𝐴 by premultiplying 𝐴 by a
certain special matrix called elementary matrices.
 Elementary Matrices

Notations:
• 𝐸𝑖𝑗 : Interchanging the 𝑖𝑡ℎ and the 𝑗𝑡ℎ row of the identity matrix 𝐼.
• 𝐸𝑖𝑗 𝑟 : Adding 𝑟 times row 𝑖 to row 𝑗 in the identity matrix 𝐼.
• 𝐸𝑖 𝑟 : Multiplying the 𝑖𝑡ℎ row of the identity matrix by the scalar 𝑟.

• Definition: The matrices 𝐸𝑖𝑗 , 𝐸𝑖𝑗 𝑟 , 𝐸𝑖 𝑟 , which are obtained by

performing elementary operations to identity matrix, are called elementary
matrices.
• Theorem: Let 𝐸 be an 𝑛 × 𝑛 elementary matrix obtained by performing a
particular elementary operation on the 𝑛 × 𝑛 identity matrix. For any 𝑛 × 𝑚
matrix 𝐴, 𝐸𝐴 is the matrix obtained by performing the same row operation on
𝐴.
 Row Echelon Form

• Definition: A row of a matrix is said to have 𝑘 leading zeros if the first k

elements of the row are all zeros and the (k + 1)th element of the row is not
zero. With this terminology, a matrix is in row echelon form if each row has
more leading zeros than the row preceding it. The first nonzero entry in
each row of a matrix in row echelon form is called a pivot. For example,
1 2 3 2 3
1 3 4
0 0 4 , 𝑎𝑛𝑑 0 6
0 1 6
0 0 0 0 0

• Definition: A row echelon matrix in which each pivot is a 1 and in which

each column containing a pivot contains no other nonzero entries is said to
be in reduced row echelon form. For example,
1 0
1 0 0
0 1 , 𝑎𝑛𝑑
0 0 0
0 0
 Row Echelon Form

• Definition: The rank of a matrix is the number of nonzero rows in its row
echelon form.
• Theorem: For any 𝑚 × 𝑛 matrix 𝐴 , there exist elementary matrices
𝐸1 , … , 𝐸𝑘 such that the matrix product 𝐸1 ⋯ 𝐸𝑘 𝐴 = 𝑈, where 𝑈 is in row
echelon form.
 Algebra of Square Matrix

• Inverse: Let 𝐴 be an 𝑛 × 𝑛 matrix . The matrix 𝐵 is an inverse for 𝐴 if

𝐴𝐵 = 𝐵𝐴 = 𝐼. If the matrix 𝐵 exists, we say that 𝐴 is invertible or
nonsingular, otherwise 𝐴 is noninvertible or singular.
• Remark:
– Inverse matrices only exist for square matrices.
– A square matrix can have at most one inverse.
• Rules for Inverse
– If 𝐴 is invertible, then 𝐴−1 −1 = 𝐴.
– If 𝐴, 𝐵 are invertible, then 𝐴𝐵 is also invertible and 𝐴𝐵 −1 = 𝐵−1 𝐴−1 .
– If 𝐴 is invertible and 𝑐 is a nonzero number, then 𝑐𝐴 is invertible and
𝑐𝐴 −1 = 𝑐 −1 𝐴−1 .
– If 𝐴 is invertible, then 𝐴𝑇 is invertible and 𝐴𝑇 −1 = 𝐴−1 𝑇 .
 Algebra of Square Matrix

• Generalized Inverse: For a matrix 𝐴𝑚×𝑛 , there is another matrix 𝐴+ 𝑛×𝑚

that satisfies the following requirements:
– 𝐴𝐴+ 𝐴 = 𝐴
– 𝐴+ 𝐴𝐴+ = 𝐴+
– 𝐴+ 𝐴 is symmetric
– 𝐴𝐴+ is symmetric
• Remark:
1. A unique 𝐴+ can be found for any matrix, whether 𝐴 is singular or not,
even if 𝐴 is not square.
2. If 𝐴 is a square nonsingular matrix, 𝐴+ = 𝐴− .
 Algebra of Square Matrix

• Powers of a Matrix: If 𝐴 is an invertible matrix, then we can take integral

powers of 𝐴. The matrix 𝐴𝑚 is defined as 𝐴 ∙ 𝐴 ⋯ 𝐴(𝑚 times). We can also
define negative powers of 𝐴 as 𝐴−𝑚 = 𝐴−1 ∙ 𝐴−1 ⋯ 𝐴−1 .
Rules for Powers:
If 𝐴 is invertible, then
– 𝐴𝑚 is invertible for any integer 𝑚 and 𝐴𝑚 −1 = 𝐴−1 𝑚 = 𝐴−𝑚 .
– For any integers 𝑟 and 𝑠, 𝐴𝑟 𝐴𝑠 = 𝐴𝑟+𝑠
• Trace: If 𝐴 is a 𝑛 × 𝑛 matrix, the race is the sum of its diagonal elements
𝑛

𝑡𝑟 𝐴 = ෍ 𝑎𝑖𝑖
𝑖=1
Rules for trace:
𝑡𝑟 𝐴 = 𝑡𝑟 𝐴′ , 𝑡𝑟 𝐴 + 𝐵 = 𝑡𝑟 𝐴 + 𝑡𝑟 𝐵 , 𝑡𝑟 𝐴𝐵 = 𝑡𝑟 𝐵𝐴
 Partitioned Matrices

• Let 𝐴 be a 𝑚 × 𝑛 matrix. A submatrix of 𝐴 is a matrix formed by

discarding some entire rows and/or columns. A partitioned matrix is a
matrix which has been partitioned into submatrix by horizontal and/or
vertical lines which extend along entire rows or columns. For example
𝑎11 𝑎12 ￤ 𝑎13 ￤ 𝑎14 𝑎15
−− −− ￤ −− ￤ −− −−
𝐴=
𝑎21 𝑎22 ￤ 𝑎23 ￤ 𝑎24 𝑎25
𝑎31 𝑎32 ￤ 𝑎33 ￤ 𝑎34 𝑎35
which we can write as
𝐴11 𝐴12 𝐴13
𝐴=
𝐴21 𝐴22 𝐴23
• Each submatrix is called a block of 𝐴.
 Addition of Partitioned Matrix

• Suppose 𝐴 and 𝐵 are two 𝑚 × 𝑛 matrix which have been partitioned in the
same way, that is
𝐴 𝐴12 𝐴13 𝐵 𝐵12 𝐵13
𝐴 = 11 , 𝐵 = 11
𝐴21 𝐴22 𝐴23 𝐵21 𝐵22 𝐵23
where 𝐴𝑖𝑗 and 𝐵𝑖𝑗 have the same dimensions. Then 𝐴 and 𝐵 can be added
as the blocks are scalars.
𝐴 + 𝐵11 𝐴12 +𝐵12 𝐴13 +𝐵13
𝐴 + 𝐵 = 11
𝐴21 +𝐵21 𝐴22 +𝐵22 𝐴23 +𝐵23
• Similarly for subtraction and scalar multiplication.
 Multiplication of Partitioned Matrix

• For two partitioned matrices 𝐴 and 𝐵, if 𝐴𝐵 exist, then block multiplication

requires the way of partitions to satisfy
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑙𝑢𝑚𝑛 𝑏𝑙𝑜𝑐𝑘𝑠 𝑜𝑓 𝐴 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑜𝑤 𝑏𝑙𝑜𝑐𝑘𝑠 𝑜𝑓 𝐵
For example,
𝐴 𝐴12 𝐵 𝐵12 𝐵13
𝐴 = 11 , 𝐵 = 11
𝐴21 𝐴22 𝐵21 𝐵22 𝐵23
Then
𝐴 𝐵 + 𝐴12 𝐵21 𝐴11 𝐵12 + 𝐴12 𝐵22 𝐴11 𝐵13 + 𝐴12 𝐵23
𝐴𝐵 = 11 11
𝐴21 𝐵11 + 𝐴22 𝐵21 𝐴21 𝐵12 + 𝐴22 𝐵22 𝐴21 𝐵13 + 𝐴22 𝐵23
For example, 𝐴 = 𝒂𝟏 , 𝒂𝟐 , each 𝒂𝑖 is 𝑚 × 1 column vector, then
𝒂′𝟏 𝒂𝟏 𝒂′𝟏 𝒂𝟐
𝐴′ 𝐴 = ′
𝒂𝟐 𝒂𝟏 𝒂′𝟐 𝒂𝟐
 Determinant

• Here we will describe a straightforward test to determine whether a square

matrix is nonsingular or not. In particular, for any square matrix we will
define a number called the determinant with the property that the square
matrix is nonsingular if and only if the determinant is nonzero.
• The determinant is defined inductively. There is a natural definition for 1 ×
1 matrices. A 1 × 1 matrix is just a scalar a.
Det 𝑎 = 𝑎
• For a 2 × 2 matrix,
𝑎11 𝑎12
𝐴= 𝑎
21 𝑎22
we define the determinant of a 2 × 2 matrix as
det 𝐴 = 𝑎11 𝑎22 − 𝑎12 𝑎21
= 𝑎11 det 𝑎22 − 𝑎12 det 𝑎21
 Determinant

• Turing to a general 𝑛 × 𝑛 matrix, let 𝐴𝑖𝑗 be the 𝑛 − 1 × 𝑛 − 1

submatrix obtained by deleting the 𝑖𝑡ℎ row and 𝑗𝑡ℎ column, 𝑀𝑖𝑗 =
det 𝐴𝑖𝑗 is called the 𝑖, 𝑗 𝑡ℎ minor of 𝐴. The determinant of 𝐴 now is
defined inductively.
• Definition
let 𝐴 be a 𝑛 × 𝑛 matrix,
For 𝑛 = 1, det 𝐴 = 𝑎11 .
For 𝑛 > 1,
det 𝐴 = 𝑎11 𝑀11 − 𝑎12 𝑀12 + ⋯ + −1 𝑛+1 𝑎1𝑛 𝑀1𝑛
𝑛

= ෍ −1 1+𝑗 𝑎 𝑀
1𝑗 1𝑗
𝑗=1
 Determinant

• Note that the 𝑎𝑖𝑗 term begins with a plus sign if 𝑖 + 𝑗 is even and with a
minus sign if 𝑖 + 𝑗 is odd. 𝐶𝑖𝑗 = −1 𝑖+𝑗 𝑀𝑖𝑗 is called the 𝑖, 𝑗 𝑡ℎ cofactor
of 𝐴. Then the determinant of 𝐴 can also be written as
𝑛

det 𝐴 = ෍ 𝑎1𝑗 𝐶1𝑗

𝑗=1
• Totally 𝑛! terms if the expression is expanded.

• Theorem: A square matrix 𝐴 is nonsingular if and only if det 𝐴 is

nonzero.
 Determinant

• Actually we can calculate the determinant of an 𝑛 × 𝑛 matrix by expanding

along any row or column.
• Let 𝐴 be an 𝑛 × 𝑛 matrix, for any 𝑖, 𝑗
𝑛

det 𝐴 = ෍ −1 𝑖+𝑘 𝑎 det 𝐴𝑖𝑘

𝑖𝑘
𝑘=1
𝑛

= ෍ −1 𝑗+𝑘 𝑎 det 𝐴𝑘𝑗

𝑘𝑗
𝑘=1
• Theorem: For any matrix 𝐴, det 𝐴 = det 𝐴𝑇 .
 Determinant

• Facts
– If 𝐵 is formed by interchanging two rows or columns of an 𝑛 × 𝑛
matrix 𝐴, then det 𝐵 = − det 𝐴 .
– If 𝐵 if formed by multiplying a scalar 𝑟 to row (or column) 𝑖, then
det 𝐵 = 𝑟 ∙ det 𝐴 .
– If two rows or columns of 𝐴 are equal, then det 𝐴 = 0.
– If matrix 𝐴 has an all-zero row (or column), then det 𝐴 = 0.
– Let 𝐴 and 𝐵 be two 𝑛 × 𝑛 matrices which differ only in their 𝑖𝑡ℎ row.
Let 𝐶 be a matrix whose the 𝑖𝑡ℎ row equals to the matrix sum of the 𝑖𝑡ℎ
rows of 𝐴 and 𝐵 and the other rows are the same with 𝐴 and B, then
det 𝐶 = det 𝐴 + det 𝐵
– Let 𝐸𝑖𝑗 𝑟 be an elementary matrix obtained by adding 𝑟 times of the
𝑖𝑡ℎ row to the 𝑗𝑡ℎ row of an identity matrix. Let 𝐴 be an 𝑛 × 𝑛 matrix,
then
det 𝐸𝑖𝑗 𝑟 𝐴 = det 𝐴
 Determinant

– Let 𝑅 be the row echelon form of 𝐴, then det 𝑅 = ± det 𝐴 .

– The determinant of a diagonal matrix 𝐴 is the product of its diagonal
entries.
– The determinant of a upper- or lower-triangular matrix is the product of
its diagonal entries.
• Theorem: Let 𝐴, 𝐵 be two arbitrary 𝑛 × 𝑛 matrix, then
det 𝐴𝐵 = det 𝐴 det 𝐵
If 𝐴 is invertible, then
det 𝐴−1 = 1Τdet 𝐴
 Adjoint of a Matrix

• Definition: For any 𝑛 × 𝑛 matrix 𝐴, let 𝐶𝑖𝑗 denotes the 𝑖, 𝑗 𝑡ℎ cofactor of

𝐴. The 𝑛 × 𝑛 matrix whose 𝑖, 𝑗 𝑡ℎ entry is 𝐶𝑗𝑖 is called the adjoint of 𝐴
and is denoted as adj 𝐴.
• Theorem: Let A be a nonsingular matrix, then
−1
1
𝐴 = adj A
det 𝐴