What is the null matrix , and how does it differ from the identity matrix?

The null matrix, also called the zero matrix , has every element equal to 0 , and it plays the part that 0 plays for ordinary numbers. Unlike the identity matrix, it does not have to be square.

When can two matrices be added or subtracted?

Only when they have the same order , meaning the same number of rows and the same number of columns. You then add or subtract the corresponding elements, position by position.

True or False? Two matrices are equal only when every pair of corresponding elements is equal.

True. Matrix equality is checked element by element , so all of the pairs have to match. That is what lets you form an ordinary equation from any one position when a matrix equation contains an unknown.

How do you square a matrix?

Write it out as the matrix multiplied by itself , then carry out the multiplication in the usual way. Do not square each element separately, and note that only square matrices can be squared at all.

Define the determinant of a matrix.

The determinant is a single number , positive or negative, worked out from the elements of a square matrix. Only square matrices have one, and its main use is in finding the inverse .

Why can a matrix with determinant zero have no inverse?

The formula divides by the determinant, and dividing by zero is not defined. Such a matrix therefore has no inverse at all.

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Properties of Matrices (Edexcel IGCSE Maths B): Flashcards

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FrontIntroduction to Matrices
Define a matrix, and say what its order means.

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Define a matrix, and say what its order means.
A matrix is a rectangular grid of elements, which may be numbers or letters, arranged in rows and columns.
Its order is its number of rows followed by its number of columns, so a matrix with 2 rows and 3 columns has order $2 \times 3$ .
What is the order of $(\begin{matrix} a & b \\ c & d \end{matrix})$ , and what is such a matrix called?
Its order is $2 \times 2$ , because it has two rows and two columns.
A matrix with the same number of rows as columns is called a square matrix.
True or False?
A $2 \times 3$ matrix has two rows and three columns.
True.
The order of a matrix is always written rows first, then columns.
So a $3 \times 2$ matrix is a different shape entirely, with three rows and two columns.
What can a $2 \times 3$ matrix be used to represent geometrically?
The three vertices of a triangle, with each vertex written as one column of the matrix.
A single column of two numbers holds one point, so three columns hold three points.
Fill in the two missing elements of the $2 \times 2$ identity matrix.
$I = (\begin{matrix} 1 & ______ \\ ______ & 1 \end{matrix})$
The completed matrix is:
$I = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$
The identity matrix has 1s down the diagonal from top left to bottom right and zeros everywhere else, and it does for matrices what the number 1 does for ordinary numbers.
What is the null matrix, and how does it differ from the identity matrix?
The null matrix, also called the zero matrix, has every element equal to 0, and it plays the part that 0 plays for ordinary numbers.
Unlike the identity matrix, it does not have to be square.
When can two matrices be added or subtracted?
Only when they have the same order, meaning the same number of rows and the same number of columns.
You then add or subtract the corresponding elements, position by position.
Fill in the two missing elements of the sum.
$(\begin{matrix} 2 & - 3 \\ 5 & 1 \end{matrix}) + (\begin{matrix} - 1 & 7 \\ - 4 & 6 \end{matrix}) = (\begin{matrix} 1 & ______ \\ ______ & 7 \end{matrix})$
The completed sum is:
$(\begin{matrix} 2 & - 3 \\ 5 & 1 \end{matrix}) + (\begin{matrix} - 1 & 7 \\ - 4 & 6 \end{matrix}) = (\begin{matrix} 1 & 4 \\ 1 & 7 \end{matrix})$
Each element of the answer comes from the two elements in the same position in the matrices being added.
How do you multiply a matrix by a scalar?
Multiply every element of the matrix by that scalar.
A scalar is simply a number, and a lower-case letter is usually used for one, so $k A$ means every element of $A$ multiplied by $k$ .
True or False?
Two matrices are equal only when every pair of corresponding elements is equal.
True.
Matrix equality is checked element by element, so all of the pairs have to match.
That is what lets you form an ordinary equation from any one position when a matrix equation contains an unknown.
Given that $2 A - n B = (\begin{matrix} 8 & 1 \\ - 13 & 2 \end{matrix})$ , how do you find $n$ ?
Work out $2 A$ and $n B$ separately, subtract them element by element, then equate any one position with the matching element on the right.
Solving that single equation gives $n$ , and the other three positions are worth checking to make sure no slip has been made.
When can two matrices be multiplied together?
Only when the number of columns in the first equals the number of rows in the second.
So a $2 \times 2$ matrix can multiply a $2 \times 3$ matrix, but a $2 \times 3$ cannot multiply a $2 \times 2$ .
Fill in the two missing entries of the product.
$(\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} ______ \\ ______ \end{matrix})$
The completed product is:
$(\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} a x + b y \\ c x + d y \end{matrix})$
Each entry pairs a row of the first matrix with the column of the second, multiplying corresponding elements and adding the results.
What is the order of the product of a $2 \times 2$ matrix and a $2 \times 3$ matrix?
It is $2 \times 3$ , taking the rows of the first and the columns of the second.
The two inner numbers have to match for the multiplication to be possible at all, and the two outer numbers give the answer's order.
True or False?
For matrices, $AB$ always gives the same result as $BA$ .
False.
Matrix multiplication is not commutative, so in general $AB \neq BA$ .
For instance $(\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}) (\begin{matrix} 0 & 1 \\ 5 & 1 \end{matrix}) = (\begin{matrix} 10 & 3 \\ 20 & 7 \end{matrix})$ , while the other order gives $(\begin{matrix} 3 & 4 \\ 8 & 14 \end{matrix})$ .
How do you square a matrix?
Write it out as the matrix multiplied by itself, then carry out the multiplication in the usual way.
Do not square each element separately, and note that only square matrices can be squared at all.
True or False?
For three matrices, $(AB) C$ gives the same result as $A (BC)$ .
True.
Matrix multiplication is associative, so which pair you multiply first makes no difference.
What must not change is the order of the letters, since $A (BC)$ is not the same as $(BC) A$ .
What happens when you multiply a matrix by the identity matrix?
Nothing changes: $AI = IA = A$ , whichever side the identity is on.
That makes the identity matrix behave like the number 1, and it is one of the few matrix products where the order genuinely does not matter.
Define the determinant of a matrix.
The determinant is a single number, positive or negative, worked out from the elements of a square matrix.
Only square matrices have one, and its main use is in finding the inverse.
Complete the formula for the determinant of a $2 \times 2$ matrix.
$\det (\begin{matrix} a & b \\ c & d \end{matrix}) =______- b c$
The completed formula is:
$\det (\begin{matrix} a & b \\ c & d \end{matrix}) = a d - b c$
It is the product of the leading diagonal, from top left to bottom right, minus the product of the other diagonal.
What is the determinant of $(\begin{matrix} 3 & 4 \\ - 2 & - 1 \end{matrix})$ ?
It is $5$ , from $3 \times (- 1) - 4 \times (- 2)$ .
That works out as $- 3 - (- 8) = - 3 + 8 = 5$ , so the double negative in the second product is the step to watch.
True or False?
A determinant can be negative.
True.
The determinant is an ordinary number and nothing forces it to be positive.
For example $\det (\begin{matrix} 2 & 4 \\ 5 & 1 \end{matrix}) = 2 - 20 = - 18$ .
What does $\det A$ mean?
It means the determinant of the matrix $A$ .
It is a number rather than a matrix, so it can be set equal to another determinant and the result solved as an ordinary equation.
Define the inverse of a matrix.
The inverse of a square matrix $A$ , written $A^{- 1}$ , is the matrix that multiplies with it to give the identity matrix $I$ .
Only square matrices can have an inverse at all.
Fill in the two missing entries of the inverse formula.
${(\begin{matrix} a & b \\ c & d \end{matrix})}^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & ______ \\ ______ & a \end{matrix})$
The completed formula is:
${(\begin{matrix} a & b \\ c & d \end{matrix})}^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix})$
Swap the two entries on the leading diagonal, change the sign of the other two, then divide the whole thing by the determinant.
True or False?
A matrix multiplied by its inverse gives the same result whichever way round they are.
True.
Both ${AA}^{- 1}$ and $A^{- 1} A$ come out as the identity matrix $I$ .
That is worth noticing, because matrix multiplication does not normally give the same answer in both orders.
Why can a matrix with determinant zero have no inverse?
The formula divides by the determinant, and dividing by zero is not defined.
Such a matrix therefore has no inverse at all.
How can you show $B$ is the inverse of $A$ without using the formula?
Multiply the two together and show that the result is the identity matrix.
Showing either $AB = I$ or $BA = I$ is enough on its own.
Find the inverse of $(\begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix})$ .
Its determinant is $1 \times 7 - 3 \times 2 = 1$ , so the fraction in front of the matrix is just 1.
Swapping the leading diagonal and changing the signs of the other two entries gives $(\begin{matrix} 7 & - 3 \\ - 2 & 1 \end{matrix})$ .
How do you solve $AB = C$ for $B$ ?
Multiply both sides from the left by $A^{- 1}$ , which gives $A^{- 1} AB = A^{- 1} C$ .
Since $A^{- 1} A$ is the identity and the identity leaves $B$ alone, this leaves $B = A^{- 1} C$ .
How do you solve $BA = C$ for $B$ ?
Multiply both sides from the right by $A^{- 1}$ , which gives ${BAA}^{- 1} = {CA}^{- 1}$ .
That leaves $B = {CA}^{- 1}$ , with the inverse written on the right of $C$ .
True or False?
When solving a matrix equation, it does not matter which side you multiply by the inverse.
False.
It matters a great deal, because matrix multiplication depends on the order.
Multiplying $BA = C$ from the left gives $A^{- 1} BA = A^{- 1} C$ , which does not simplify any further.
Why does $A^{- 1} BA$ not simplify?
A matrix and its inverse only collapse to the identity when they are next to each other in the multiplication.
Here $B$ sits between them, so neither pairing can be made and nothing cancels.
Fill in the missing step of this simplification.
$A^{- 1} AB =______= B$
The completed simplification is:
$A^{- 1} AB = IB = B$
$A^{- 1} A$ collapses to the identity $I$ , and multiplying by the identity leaves $B$ unchanged.

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